THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


!^ 


r-=H   he'- 


UNIVERSITY  of  CALIFORNl 
AT 
LOS  ANGELES 
LIBRARY 


asiter^iDe  €Ducational  jmonograpljjs 

EDITED  BY  HENRY  SUZZALLO 

PRESIDENT   OF   THE    UNIVERSITY   OF    WASHINGTON 
SEATTLE,    WASHINGTON 

THE  TEACHING  OF 
HIGH  SCHOOL 
MATHEMATICS 

BY 
GEORGE  W.  EVANS 

HEADMASTER   OF   THE   CHARLESTOWN   HIGH   SCHOOL 


HOUGHTON    MIFFLIN    COMPANY 

BOSTON      NEW  YORK      CHICAGO     SAN  FRANCISCO 


COPYRIGHT,    59II,   BY   HOUGHTON    MIFFLIN   COMPAWV 
ALL  RIGHTS   RESERVED 


QTijt  Bibtrsitit  JOnii 

CAMBRIDGE  .  MASSACHUSETTS 
U    .    S   .    A 


r  ■    ^- 


^  CONTENTS 

to 

Editor's  Introduction     ...  v 

I.   The  Modern  Point  of  View      .  .       i 

11.   The  Order  of  Topics      ...  13 

III.   Equations  and  their  Use  .        .  •     17 

r4       IV.   Some  Rules  of  Thumb     ...  23 

'^j,,^^^      V.   Geometry  as  Algebraic  Material  .    32 

*\       VI.   The  Graphical  Method  ...  41 

VII.   The  Bases  of  Proof  in  Geometry  .     49 

VIII.   The  Method  of  Limits   ...  60 

^        IX.    Simpson's   Rule   and   the   Curve   of 

Sections 74 

X.  The  Teacher 85 

Outline 91 


EDITOR'S    INTRODUCTION 

The  ideal  of  practicality  has  now  entered  the 
schools  with  telling  force.  It  has  been  manifested 
in  its  demand  for  vocational  training,  and  it  is 
reconstructing  the  older  cultural  training  by  elim- 
inations and  additions.  Its  effects  on  the  curri- 
cula of  liberal  schools  are  quite  obvious.  Materials 
once  accepted  without  question,  when  schools 
had  a  margin  of  energy,  are  now  displaced  by  the 
pressure  of  new  demands.  These  new  demands 
force  the  teachers  of  a  crowded  curriculum  to  re- 
consider every  traditionally  taught  fact  or  process 
from  the  standpoint  of  its  relative  rather  than  its 
absolute  worth.  Without  scorn  for  the  value  of 
any  truth,  old  disciplines  disappear  and  new  ones 
enter.  The  ends  of  life  are  the  ends  of  the  school, 
and  what  is  social  is  becoming  academic,  thus 
freeing  the  scholastic  from  the  contempt  of  men 
who  work  in  the  world  at  large. 

What  the  practical  ideal  is  doing  for  the  pur- 
poses of  the  school,  the  ideal  of  psychological 


EDITOR'S  INTRODUCTION 

efficiency  is  doing  for  its  methods.  It  is  perceived 
that  truths  that  stand  in  vital  relation  to  human 
need  are  more  readily  mastered  and  retained  than 
those  that  do  not ;  that  a  series  of  facts  is  better 
presented  in  an  order  that  pays  heed  to  the  stu- 
dent's mental  development  rather  than  to  the 
logical  steps  in  a  perfected  system  of  knowledge. 
Hence,  old  approaches  to  the  mastery  of  studies 
are  giving  way  to  more  objective,  more  reasonable, 
and  more  circuitous,  but  more  efficient  means  of 
progression. 

It  is  interesting  to  note  how  often  the  demands 
for  more  practical  results  and  more  efficient  psy- 
chological methods  cooperate  toward  a  common 
reformation  of  school  work.  The  practicalists  tell 
us  that  the  schools  must  equip  men  with  what 
they  will  most  use  as  citizens  and  as  workers. 
The  psychologists  say  it  is  folly  to  master  those 
minor  knowledges  and  disciplines,  which  life  will 
not  utilize,  for,  neglected,  the  memory  of  them 
soon  fades,  and  the  school  will  have  had  only  its 
pains  for  its  trouble.  The  world  asks  for  live  men 
who  know  the  conditions  of  real  life  in  the  com- 
bat as  they  find  it,  men  who  can  think  through 
vi 


EDITOR'S   INTRODUCTION 

a  new  situation,  when  old  knowledge  fails  to 
point  the  way.  And,  for  reasons  quite  his  own, 
the  skilled  teacher,  who  knows  human  nature, 
prefers  to  let  his  pupil  think  slowly  through  his 
difficulties,  far  more  solicitous  that  the  child 
get  a  confident  command  over  his  own  mental 
resources  than  that  he  accumulate  many  facts 
quickly  by  the  direct  method  of  memorization. 
Psychology  and  practicality  are  the  twin  reform- 
ers of  the  schools,  and  like  true  twins  they  man- 
ifest a  deal  of  similarity  in  their  manner  of  ex- 
pressing themselves  in  actual  practice. 

The  modern  demands  of  a  sound  educational 
sociology  and  a  rational  educational  psychology 
do  not  appear  so  revolutionary  in  the  elementary 
school  as  in  the  secondary  schools ;  for  element- 
ary teachers  have  become  more  used  to  revolu- 
tion, and  the  violations  of  professional  habit  do 
not  seem  so  monstrous  where,  as  in  the  element- 
ary school,  the  sanctuaries  of  tradition  are  not 
so  ancient.  The  secondary  school  teacher  is  also 
far  more  of  a  specialist  and  less  open  to  the  con- 
sideration of  the  total  effects  of  a  youth's  edu- 
cation; he  is  a  little  nearer  to  the  university 
vii 


EDITOR'S   INTRODUCTION 

where  truth  for  its  own  sake  is  a  dominant  ideal. 
In  consequence,  he  suffers  a  considerable  wrench 
when  the  logical  arrangements  of  systematic 
knowledge  arc  disturbed  for  the  purposes  of  more 
vital  presentation.  The  modern  programme  of 
reform  therefore  finds  a  less  easy  acceptance  in 
the  secondary  school. 

But  high  school  teachers,  just  because  they  are 
specialized  in  interest  and  responsibility,  are  not 
all  of  one  pattern.  They  are  likely  to  resist  change 
in  varying  degrees.  One  can  readily  see  that  in- 
structors in  history  and  in  science  could  be  more 
easily  led  to  reconstruct  their  courses  of  study 
so  as  to  interpret  present  problems  than  could 
teachers  of  the  classics  or  of  mathematics.  These 
last  named  subjects  have  had  a  very  old  and  hon- 
orable position  in  the  curriculum.  The  classics, 
by  virtue  of  their  antiquity,  could  not  stand  much 
modernization  without  losing  their  essence;  and 
advanced  mathematics,  the  most  abstract  and 
formal  of  all  the  high  school  disciplines,  lends 
itself  to  concreteness  and  practical  application 
only  with  great  effort.  Modify  the  logical  se- 
quence of  mathematics  at  a  single  point,  and 
viii 


EDITOR'S   INTRODUCTION 

you  disturb  the  whole  structure.  A  single  mino? 
change  that  has  a  far-reaching  effect  is  therefore 
likely  to  be  resisted. 

But  traditional  resistance  of  an  heroic  sort  may 
finally  be  futile.  Sound  sanctions  at  last  decide 
every  controversy.  What  is  right  must  prevail 
in  the  teaching  of  mathematics,  as  in  morals  or 
politics.  With  such  faith  in  mind,  this  volume 
on  the  teaching  of  high  school  mathematics  is 
given  to  the  teaching  profession.  A  reflection  of 
modern  demands  upon  the  school  and  an  expres- 
sion of  recent  pedagogical  ideals,  it  summarizes 
the  modern  reform  programme  as  applied  to 
mathematical  teaching.  It  aims  in  considerable 
degree  to  make  mathematics  yield  practical  ef- 
ficiency, and  it  derives  from  the  experiences  of 
the  students  themselves  the  impulse  and  power 
to  use  high  school  mathematics  as  an  instrument 
for  the  solution  of  real  needs. 

The  chaotic  condition  in  which  the  discussions 
of  the  past  decade  have  left  the  subject  of  math- 
ematical teaching  suggests  the  desirability  of  pre- 
senting, in  small  compass,  a  systematic  restate- 
ment, not  merely  in  terms  of  a  general  theory, 
ix 


EDITOR'S   INTRODUCTION 

but  also  in  the  more  useful  form  of  a  series  of 
concrete  suggestions  as  to  the  material  and  meth- 
ods to  be  used.  This  volume  is  offered  with  the 
assurance  that  it  serves  this  definite  purpose. 


THE  TEACHING  OF 
HIGH  SCHOOL  MATHEMATICS 

I 

THE  MODERN  POINT  OF  VIEW 

The  purpose  of  most  of  the  recent  innovations 
in  the  teaching  of  high  school  mathematics  is  to 
provide  a  more  immediate  application  of  the 
knowledge  acquired;  and  to  make  the  success- 
ive steps  of  the  pupil's  progress  available  even 
if  his  education  is  interrupted  before  the  com- 
pletion of  the  high  school  course. 

It  is  to  be  remembered  at  the  start  that  the  pu- 
pils of  high  schools,  the  country  over,  do  not  for 
the  most  part  go  to  any  other  educational  insti- 
tution upon  graduation ;  that  only  a  minority 
graduate  at  all ;  and  that  a  very  large  part  of  the 
entering  class  does  not  complete  the  first  year. 
Let  us  grant,  then,  that  it  is  worth  while  for 
these  schools  to  furnish  preparation  for  college 
or  technical  schools  to  the  very  small  fraction  of 
their  membership  that  can  make  use  of  this  priv- 
I 


TEACHING   MATHEMATICS 

ilege  ;  but  we  cannot  avoid  the  conclusion  that  it 
is  no  less  important  to  give  to  those  pupils  who 
cannot  stay  through,  or  cannot  go  farther,  some 
definite  advantage  even  from  the  curtailed  study 
that  they  can  give  to  the  subject.  This  is  the 
justification  of  the  innovations  to  which  I  have 
referred. 

In  some  schools  a  six-year  secondary  school 
course  is  given,  beginning  with  the  seventh  year 
of  the  pupil's  school  life ;  and  it  is  altogether 
probable  that  this  plan  will  be  largely  extended. 
The  obstacle  to  it  is  the  insufficient  training 
obtainable  for  the  teachers  of  the  seventh  and 
eighth  years  of  the  elementary  schools,  and  the 
lack  of  departmental  organization  for  those  years. 
For  this  last  reason  teachers  who  are  excellent 
in  one  line  of  work  must  give  a  part  of  their  time 
to  the  teaching  of  subjects  for  which  they  have 
less  taste  and  aptitude  —  and  less  skill.  Mathe- 
matics suffers  more  than  any  other  subject  in 
this  respect.  In  spite  of  this  obstacle  the  plan 
is  being  urgently  advocated,  especially  in  New 
York,  and  devices  of  reorganization  have  been 
proposed  that  seem  feasible. 

2 


THE  MODERN  POINT   OF   VIEW     • 

Most  secondary  schools,  however,  receive  pu- 
pils at  the  end  of  the  eighth  year,  practically 
ignorant  of  algebra  and  geometry,  and  with  no 
confidence  whatever  in  their  own  ability  to  per- 
form the  computations  which  they  have  been 
studying  and  practicing  for  years.  The  high 
school  teacher's  problem  is  to  furnish  to  these 
pupils  instruction  that  will  immediately  improve 
their  efficiency  and  at  the  same  time  so  contri- 
bute to  their  progress  in  mathematical  know- 
ledge that  at  the  end  of  four  years  they  can  take 
up  college  work.  The  problem  is  not  yet  com- 
pletely solved,  but  much  progress  has  been  made. 
When  it  is  completely  solved  a  new  problem  may 
be  hoped  for. 

On  the  new  programme  of  high  school  mathe- 
matics the  first  thing  is  the  treatment  of  compu- 
tation—  of  the  four  operations  in  arithmetic  — 
with  emphasis  on  self-reliance  and  on  accuracy. 
Commercial  computations  may  be  left  to  the 
commercial  courses,  but  accuracy  and  self-reli- 
ance can  be  cultivated  in  such  computations  as 
more  intelligent  artisans  use.  By  this  I  mean 
only  that  in  the  successive  development  of  the 
3 


TEACHING  MATHEMATICS 

different  subjects  of  instruction  the  problems 
be  related  to  things  that  are  actually  done  in  the 
world,  —  let  us  say  in  play,  as  well  as  in  work, 

not   by    any  means  that   the    mathematical 

teacher  should  give  an  exhaustive  discussion  of 
trade  problems  for  their  own  sake.  It  is  out  of 
the  question  that  the  multitudinous  activities  of 
life  can  be  so  prepared  for  that  every  pupil  on 
leaving  school  shall  find  his  perplexities  all  solv- 
able as  corollaries  of  his  school  problems — but 
it  is  not  out  of  the  question  that  his  school  pro- 
blems should  be  such  as  men  in  the  world  about 
him  have  solved  and  are  solving  for  the  daily 
needs  of  civilization. 

The  most  obvious  points  of  contact  between 
mathematical  science  and  practical  matters  are 
the  graphical  method  and  the  use  of  formulae. 
The  graphical  method  can  be  used  from  the 
start,  even  in  the  elementary  school,  and  through- 
out the  high  school  course  with  increasing  ad- 
vantage ;  and  the  use  of  formulae  can  serve  to 
enliven  the  practice  in  arithmetic  on  the  one 
hand,  and  on  the  other  to  introduce  the  subject 
of  algebra.  In  these  two  things  the  pupil  has  his 
4 


THE  MODERN  POINT  OF  VIEW 

introduction  to  algebra  and  geometry,  as  rein- 
forcements of  his  old  enemy,  arithmetic.  To  his 
father's  question,  "What  were  you  doing  in 
school  to-day?"  he  will  reply,  "Mathematics. 
Something  like  arithmetic. "  Let  us  hope  he  will 
add  that  it  is  more  interesting. 

It  is  doubtful  if  there  is  any  advantage  at  this 
stage  in  definitions  of  algebra,  geometry,  or 
mathematics  —  or  even  of  arithmetic — if  they 
can  be  avoided.  The  pupil's  own  definition  of 
the  study  that  he  has  known  in  the  elementary 
school  as  arithmetic  would  be  interesting  if  he 
would  talk  sense,  instead  of  trying  to  say  what 
he  thinks  is  expected.  His  own  definition  of  al- 
gebra and  geometry  on  such  slight  acquaintance 
would  be  as  futile  as  Dickens's  impressions  of 
this  country,  recorded  in  the  "American  Notes." 
Clearness  and  definiteness  of  aim  in  teaching 
requires,  however,  that  the  teacher  should  have 
in  mind  definitions  of  his  subjects  from  the  high 
school  point  of  view.  For  this  reason  the  follow- 
ing definitions  are  selected. 

Mathematics  is  the  science  that  draws  neces- 
sary conclusions. 

5 


TEACHING  MATHEMATICS 

Algebra  is  a  systematic  method  of  abbreviat- 
ing the  words  used  in  discussing  numbers. 

Geometry  is  a  method  of  investigating  the 
shape  and  size  of  material  things  by  means 
of  diagrams,  and  of  expressing  the  relations 
of  shape  and  size  among  these  diagrams  by 
means  of  numbers. 

The  word  mathematics  comes  from  a  Greek 
word  meaning  to  know,  and  was  originally  used 
for  science  in  general,  becoming  ls>.ter  restricted 
to  numbers,  geometry,  and  kindred  subjects; 
and  at  one  time  even  to  astronomy.  The  com- 
prehensive definition  here  given  is  quite  mod- 
ern ;  the  fact  that  it  includes  logic  gives  a  valu- 
able point  of  view  to  the  teacher  of  geometry. 

The  word  geometry  indicates  its  origin  in  the 
barter  of  land.  Abraham's  purchase  of  the  field 
of  Machpelah,  notwithstanding  the  friendly  dis- 
claimer of  Ephron  the  Hittite  ("What  is  that 
betwixt  me  and  thee  .-'  "  )  shows  us  a  conviction, 
even  among  the  kindly  children  of  Heth,  that 
equitable  dealing  was  better  than  kind  hearts. 
At  any  rate  their  neighbors  the  Egyptians, 
cultivating  lands  on  which  the  landmarks  were 
periodically  washed  away,  found  it  necessary 
6 


THE  MODERN   POINT  OF  VIEW 

to  locate  and  measure  their  holdings.  From 
this  utilitarian  practice  the  Greeks  derived 
their  bases  for  a  science  which  they  felt  to  be 
abstract  and  ideal,  and  which  has  come  down 
to  our  own  day,  only  slightly  changed  in  ma- 
terial and  method,  as  a  subject  for  high  school 
teaching. 

With  the  Greeks  it  was  a  study  of  the  relative 
shapes  and  sizes  of  ideal  diagrams,  mostly  plane  ; 
but  it  was  also  a  study  of  certain  relations  of  num- 
bers such  as  need  not  be  obtained  by  counting  ; 
these  numbers  being  represented  by  lines,  and 
their  relations  by  the  relations  of  the  lines  in  dia- 
grams. With  the  notation  of  modern  algebra,  the 
numerical  part  of  the  Greek  geometry  has  been 
superseded  by  the  study  of  formulas ;  and  there 
has  arisen  a  superstitious  belief  in  the  necessity 
for  the  absolute  separation  of  geometry  and  alge- 
bra, which  has  materially  impeded,  for  more  than 
a  century,  the  natural  development  of  teaching  in 
mathematics.  For  our  purposes,  geometry  is  not 
only  a  numerical  investigation  of  the  ideas  of 
space,  but  a  diagrammatic  representation  of 
number.  A  chief  purpose  must  be  to  correlate 
7 


TEACHING  MATHEMATICS 

the  ideas  of  number  and  space  so  as  to  enlarge 
the  idea  of  number  by  that  association. 

To  be  sure  there  are  important  properties  and 
relations  of  our  diagrams  that  are  not  numerical, 
and  the  investigation  of  them  is  by  no  means  an 
unimportant  part  of  our  study.  Again,  we  use 
geometrical  arguments  as  a  type  of  valid  in- 
ference ;  by  them  we  teach  our  pupils  to  avoid 
fallacies  and  to  examine  demonstrations  with 
confidence  in  their  own  judgment.  Finally, 
from  the  beginning  of  his  study  of  geometry,  the 
pupil  is  learning  to  comprehend  the  precise 
meaning  of  definite  statements  about  simple 
things,  and  to  separate  from  his  knowledge  of  a 
particular  object  the  connotations  that  are  not 
postulated  as  a  part  of  the  basis  of  his  argument. 
In  all  these  respects  the  subject  which  we  call 
geometry  differs  notably  from  land-measurement, 
which  the  term  originally  implied ;  but  the 
measurement  of  land  and  of  plane  areas  gener- 
ally furnishes  for  our  childhood,  as  it  did  for 
the  childhood  of  science,  an  excellent  introduc- 
tion and  point  of  departure. 

The  word  algebra  is  the  Arabic  name  of  a 
8 


THE  MODERN  POINT  OF  VIEW 

single  but  important  detail  in  the  rule  that  the 
Arabian  algebraists  used  for  the  reduction  of 
simple  equations.  Their  rule  was  first  to  add  to 
each  member  of  the  equation  what  would  get  rid 
of  all  negative  terms  ;  then  to  subtract  from 
each  side  what  would  leave  the  unknown  term 
standing  alone.  To  the  first  of  these  two  opera- 
tions was  applied  the  Arabic  word  from  which 
algebra  is  derived.  We  may  conclude  then  that 
so  far  as  the  derivation  of  its  name  may  in- 
dicate, algebra  was  originally  concerned  mainly 
with  equations  ;  and  that  is  not  a  bad  notion 
to  have  in  mind  in  planning  the  first  year  of 
high  school  mathematics. 

To  the  pupil  beginning,  however,  it  is  neces- 
sary to  show  why  equations  should  exist.  It  is 
unwise  to  take  ready-made  equations  to  show  him, 
giving  him  successively  greater  degrees  of  com- 
plication in  their  structure  and  manipulation, 
bidding  him  to  have  faith  in  the  overruling 
wisdom  of  those  who  plan  his  studies  for  bene- 
ficent ends,  and  reserving  the  problems  which 
may  give  rise  to  the  equations  until  the  equation 
itself  is  a  perfected  and  more  or  less  mechanical 
9 


TEACHING  MATHEMATICS 

instrument  in  his  hands.  On  the  contrary,  the 
equation  must  appear  to  him  at  the  begin- 
ning as  a  convenience,  if  not  a  necessity,  in  the 
discussion  of  things  that  the  pupil  may  see 
a  reason  for  discussing ;  that  is,  as  a  statement 
of  one  item  in  a  scheduled  explanation  of  a 
problem. 

Now  the  explanations  of  such  problems  as 
are  traditional  in  the  teaching  of  algebra  were 
originally  written  out  in  words,  —  what  Nessel- 
man  calls  rhetorical  algebra  ;  then  a  good  many 
systematic  abbreviations  were  used  for  the  num- 
bers involved  and  for  their  relations,  though 
the  form  of  exposition  was  still  that  of  ordinary 
speech  ;  finally  a  system  of  notation  was  devised 
which  had  no  necessary  connection  with  the 
particular  words,  and  which  did  not  follow  the 
rules  of  speech.  This  is  the  modern,  or  sym- 
bolic,  form.  It  is  capable  of  translation  into 
words,  at  least  for  the  simple  equations  that  are 
used  in  first  year  work ;  and  the  words  of  the 
problems  that  present  themselves  for  solution  by 
algebraic  means  must  always  be  capable  of  being 
so  recast  that  the  necessary  data  of  the  problem 
lo 


THE  MODERN  POINT  OF  VIEW 

can  be  "  translated  into  algebra, "  that  is,  ex- 
pressed as  an  algebraic  equation.  It  is  perfectly 
consistent  for  the  teacher,  then,  and  reasonably 
clear  for  the  pupil,  to  define  algebra  as  a  system- 
atic method  of  abbreviating  the  words  used 
in  discussing  numbers  ;  though  the  teacher 
knows  that  in  order  to  utilize  this  method  it  is 
necessary  to  rearrange  and  schematize  the  suc- 
cessive steps  of  the  discussion. 

This  definition  has  the  further  advantage  that 
it  includes  the  use  of  algebra  for  formulae,  as  a 
substitute  for  verbal  rules  of  computation.  The 
words  describing  the  numerical  data,  and  the 
operations  which  the  rule  directs  to  be  performed 
on  them,  are  all  represented  by  a  systematic 
symbolism. 

There  is  no  need  to  guard  this  definition  of 
algebra  on  account  of  any  scruples  the  teacher 
may  have  as  to  the  existence  of  purely  symbolic 
equations  and  expressions.  As  the  pupil  pro- 
gresses in  his  work  he  will  gradually  introduce 
himself  to  symbolic  manipulation  which  he  does 
not  care  to  translate  into  words, — at  least,  until 
he  arrives  at  a  rather  simple  result ;  and  there 
II 


TEACHING  MATHEMATICS 

will  be  no  awkwardness  in  his  progress  towards 
a  state  of  mind  in  which  he  will  complacently 
put  a  problem  in  at  one  end  of  an  algebraic  solu- 
tion, and  machine  out  his  answer  at  the  other, 
with  the  comfortable  assurance  that  his  algebra 
has  served  him  rather  as  a  substitute  for  thought 
than  as  a  means  for  briefly  writing  his  thought 
down. 

This  is  no  mere  figure  of  speech,  for  machines 
have  been  constructed  for  performing  highly 
complicated  algebraic  work,  and  even  for  deduc- 
ing, by  rigid  logical  rules,  simple  and  necessary 
conclusions  from  data  so  complex  that  an  un- 
trained thinker  might  easily  fail  to  make  a  com- 
plete deduction.  All  this  the  pupil  would  not 
understand  at  the  start,  and  it  is  better  that  he 
should  not ;  but  as  his  progress  continues,  these 
things  shall  be  added  unto  him. 


n 

THE  ORDER  OF  TOPICS 

It  is  a  bad  habit  of  teachers  of  all  kinds  of  sub- 
jects that  they  hunger  and  thirst  after  thorough- 
ness ;  not,  alas !  thoroughness  on  the  pupil's 
part,  but  thoroughness  of  exposition  on  the 
teacher's  part.  Thus  makers  of  text-books,  find- 
ing that  square  root  may  be  of  use  in  solving 
quadratic  equations,  exploit  not  only  the  subject 
of  square  root  of  numbers  and  of  literal  express- 
ions, but  also  the  subject  of  radicals,  and  com- 
pile sets  of  examples  under  these  heads  until  not 
only  the  pupils,  but  probably  the  teachers  also, 
forget  how  little  the  material  serves  for  handling 
equations. 

For  the  sake  of  unity,  the  first  year  student 
should  be  spared  all  such  elaborate  treatment; 
his  attention  should  be  confined  to  two  things : 
first,  the  association  of  number  with  magnitude, 
and  the  consequent  representation  of  number  by 
diagrams  ;  and  second,  the  use  of  the  equation 
13 


TEACHING  MATHEMATICS 

as  a  means  of  solving  problems.  These  two  things 
can  be  closely  related  by  using  the  formulae  of 
geometry  as  a  source  of  some  of  the  equations 
upon  which  the  student  practises  his  growing 
skill. 

The  manner  in  which  these  desiderata  are 
attained  will  probably  be  worked  out  by  each 
teacher  for  himself.  The  following  order  of  topics 
is,  consequently,  merely  suggestive. 

I.  Problems  giving  rise  to  simple  equations 
of  gradually  increasing  complication. 
II.  The  use  of  formulae  illustrated  by  the  men- 
suration of  plane  figures. 

III.  Computation,  with  economy  in  the  number 
of  figures  used  ;  square  root  of  numbers. 

IV.  Simple  transformations,  including  frac- 
tions ;  long  division  and  multiplication  with 
binomial  factors  only  ;  the  parenthesis  and 
the  radical  sign  as  symbols  of  convenience, 
not  as  sources  of  problems. 

V.  Algebraic  theorems,  illustrated  by  the  dif- 
ference of  two  squares,  the  square  of  the 
sum,  and  the  square  of  the  difference,  with 
illustrations  from  arithmetic  and  from  ge- 
ometry. Factoring  of  expressions  like  x^- 
5;r-i4  by  inspection,  and  of  expressions 
like  jir2-ioo.:r  -|-  2491  by  "completing  the 
square." 

14 


THE  ORDER  OF  TOPICS 

VI.  Quadratic  equations. 
VII.   Similar   triangles    and    the    Pythagorean 
theorem  as  a  source  of  quadratic  equations. 
VIII.  The  Graphical  Method  ;  the  straight  line 
as  the  locus  of  a  two-letter  equation  of  the 
first  degree. 
IX.  Elimination  "by  addition  or  subtraction " 
for  two-letter  equations  of  the  first  degree  ; 
rough  check  by  intersection  of  loci. 
X.  Elimination  "  by  substitution  "  for  linear- 
quadratic  pairs.   The  equation  of  a  circle 
with  given  centre  and  radius.  The  stand- 
ard parabola  j  =  ;tr2.    Graphic  checking  of 
certain  equation-pairs  and  of   a  few  one- 
letter  quadratics. 


This  will  complete  the  first  year  work.  With 
the  second  year  the  pupil  will  have  two  things  to 
do  :  to  investigate  the  methods  of  manipulating 
algebraic  expressions,  and  to  make  a  beginning 
of  the  study  of  geometry  as  a  logical  structure. 
Some  of  the  topics  under  these  heads,  such  as 
the  binomial  theorem,  or  the  method  of  limits, 
might  with  great  advantage  be  postponed  in 
favor  of  the  simpler  topics  in  solid  geometry  and 
in  trigonometry.  Those  that  are  taken  will  be  the 
IS 


TEACHING  MATHEMATICS 

more  welcome  as  the  pupil  now  realizes  that 
algebraic  expressions  have  been  of  use  to  him, 
and  as  he  has  had  experience  in  the  intelligent 
discussion  of  geometrical  inferences. 


Ill 

EQUATIONS  AND  THEIR  USE 

For  the  points  of  beginning  in  high  school  math- 
ematics we  have  problems  that  give  rise  to  alge- 
braic equations,  and  the  numerical  investiga- 
tion of  geometrical  diagrams.  The  ends  to  be 
sought  are,  in  the  first  place,  a  conscientious 
logic  and  a  sense  of  responsibility  for  numer- 
ical results ;  eventually,  the  power  to  repre- 
sent numerical  results  by  diagrams,  and  to  inter- 
pret the  relations  of  diagrams  numerically;  and 
incidentally,  facility  in  the  manipulation  of  sym- 
bolic statements  that  represent  or  replace  argu- 
ment. 

"  Real  applied  problems  "  in  algebra  for  first- 
year  high  school  pupils  are  scarce.  Arithmetic  in 
the  elementary  school  has  been  mainly  commer- 
cial, and  reflection  will  show  that  every  day 
affairs  do  not  present  many  problems  in  which 
algebraic  treatment  is  imperative.  It  is  not  expe- 
dient to  give  explanations  of  trades  or  of  science 
17 


TEACHING   MATHEMATICS 

hitherto  unknown  to  the  pupil,  for  the  sake  of 
new  problem  material. 

The  bulk  of  the  subject-matter,  then,  in  the 
problems  of  the  first  part  of  algebra,  must  be 
money,  percentage,  distances,  lapse  of  time,  and 
weights.  The  idea  of  ratio  must  be  used  from  the 
first ;  it  is  usually  wrapped  in  confusing  words, 
like  the  mist  in  which  yEneas  walked  in  the 
sunny  square  at  Carthage.  If  ratio  is  defined 
plainly  as  a  multiplier  (or  as  a  quotient)  and  is 
not  followed  by  the  usual  generalized  treatment 
of  proportion,  experience  shows  that  pupils  have 
no  difificulty  with  it. 

The  equations  to  which  these  problems  are  to 
give  rise  must  be  at  first  extremely  simple ;  so 
simple  in  fact  that  it  cannot  be  claimed  that 
their  use  makes  the  problem  one  whit  easier  to 
solve.  The  motive  to  be  presented  for  using  alge- 
bra is  that  by  its  use  the  explanation  of  the  pro- 
blem can  be  systematized  and  briefly  recorded. 
This  system  and  this  brevity  can  afterwards  be 
applied  in  the  study  of  problems  so  much  more 
complicated  that  the  pupil  trying  to  solve  them 
without  algebra  would  surely  become  confused, 
i8 


EQUATIONS  AND  THEIR  USE 

These  introductory  problems,  again,  must  be 
brief  in  statement,  and  unmistakable  in  meaning. 
New  words,  when  introduced,  must  come  one  at 
a  time,  with  full  explanation  and  illustration,  but 
not  necessarily  with  formal  definition.  The  data, 
clad  in  terms  of  money,  distance,  time,  etc., 
should  be  as  follows:  — 

First,  one  of  two  numbers,  and  its  ratio  to  an- 
other. 

Second,  the  ratio  of  two  numbers,  and  their 
sum. 

Third,  the  difference  of  two  numbers,  and 
their  sum. 

Thus  we  introduce  simple  equations  in  which 
only  positive  terms  appear.  After  the  pupil  has 
learned  how  to  deal  with  small  numbers,  attention 
can  be  focused  on  the  processes  of  reduction 
by  using  large  numbers.  The  range  of  material 
can  be  enlarged  by  introducing  the  measurement 
of  angles,  with  a  protractor,  in  degrees  and 
decimals  of  a  degree  ;  the  words  complement 
and  supplement  can  be  used  ;  and  the  number  of 
degrees  in  an  arc  distinguished  from  the  length 
of  the  arc  in  inches  or  feet. 

In  the  next  type  of  problem  negative  terms 
19 


TEACHING  MATHEMATICS 

are  introduced.  These  cannot  be  avoided,  for 
example,  when  the  sum  of  two  numbers  is  given 
and  an  equation  formed  with  multiples  of  them. 
Thus,  in  the  problem  :  — 

Two  adjacent  building  lots  have  a  total  front- 
age of  104  feet;  one  is  50  feet  deep,  the  other 
80,  and  their  areas  are  equal.  Find  the  frontage 
of  each  lot. 

SOX  =  80  (104— Jf) 

For  the  purpose  of  dealing  with  this  negative 
term  it  is  not  desirable  to  go  into  a  general  treat- 
ment of  negative  number  as  a  new  feature  of 
algebra.  The  pupil  as  yet  has  no  need  of  the  idea 
of  negativ^e  number,  except  as  the  subtrahend 
which  he  has  always  considered  it ;  he  will  not 
have  need  of  the  idea  until  he  gets  to  the  nega- 
tive solution  of  a  quadratic  equation. 

The  method  of  dealing  with  the  negative  term 
in  the  equation 

^ox  =  8320— 8o;r 
is  indicated  by  the  original  meaning  of  the  word 
"algebra,"  referred  to  in  the  first  chapter;  that 
is,  "the  making  up  of  shortages."  The  second 
member  of  the  equation,  if  it  were  not  for  the 
20 


EQUATIONS  AND  THEIR  USE 

term  — 8o;tr,  would  be  8320;  it  is  now  short  of 
8320,  and  the  shortage  is  Sox.  We  can  make  up 
that  shortage  by  adding  Sox  to  each  side  of  the 
equation,  and  the  rest  of  the  work  is  clear  to  the 
pupil. 

If  this  plan  is  adopted,  the  teacher  must  get 
rid  of  his  foolish  partiality  for  the  left  side  of  the 
equation,  and  be  ready  to  have  his  ;r-terms  on 
the  right  if  they  come  that  way.  Thus, 

65— -f  =  4  X 
becomes 

65  =  5^ 
by  adding  x  to  each  side  ;  and 

13   =  X 
is  then  just  as  good  as 

X  =  IS 
Another  advance  in  manipulation  is  necessary 
for  the  multiplication 

80  (104— ;r)    . 

and  this  should  be  dealt  with  here,  explained, 
illustrated,  emphasized,  and  practiced  upon,  as  if 
that  was  the  only  multiplication  difficulty  in 
algebra. 

21 


TEACHING  MATHEMATICS 

The  geometric  material  can  be  increased  by 
the  idea  of  a  stripe  (the  figure  formed  by  two 
parallel  lines)  and  by  the  sum  of  the  angles  of  a 
triangle.  Single  capital  letters  can  be  used  for 
the  number  of  degrees  in  an  angle,  and  the  va- 
rious theorems  about  the  angles  made  by  a 
transversal  of  two  parallels  should  be  condensed 
into  a  single  sentence,  such  as:  "All  the  acute 
angles  formed  by  any  transversal  of  a  stripe  are 
equal." 


SOME  RULES  OF  THUMB 

The  first  year  of  high  school  work  is  the  place 
to  introduce  certain  features  of  computation 
which  are  important  to  practical  men  as  well  as 
to  mathematicians.  The  chief  of  these  is  the 
measure  of  precision  in  a  measurement-number, 
as  indicated  by  the  number  of  significant  figures 
in  it. 

This  may  be  illustrated  by  the  mean  distance 
of  the  earth  from  the  sun,  which  is  about  92 J4 
million  miles.  This  may  be  a  hundred  thousand 
miles  or  more  out  of  the  way.  If  written  92,500,- 
000  the  first  three  figures  would  be  significant, 
the  others  merely  fix  the  position  of  the  decimal 
point.  Again,  the  United  States  Government 
defines  the  inch  by  saying  that  39.370  inches 
make  a  meter.  Here  the  o  is  significant,  because 
it  indicates  that  this  specification  is  accurate  to 
five  figures.  Still  another  illustration  is  the 
length  of  a  micron,  a  unit  that  may  be  used  to 
23 


TEACHING   MATHEMATICS 

measure  in  the  field  of  a  microscope.  A  micron 
is  .000039370  inches  long.  Here  the  four  zeros 
are  not  significant,  because  they  serve  merely  to 
fix  the  position  of  the  decimal  point;  the  last 
zero  is  significant. 

A  change  in  the  unit  of  measurement  will 
change  the  position  of  the  decimal  point,  but 
will  not  change  the  degree  of  accuracy  to  which  a 
measurement  is  made.  Thus  a  distance  of  1.3420 
miles  would  really  be  measured  in  feet,  and 
the  measurement  number  7085.8  for  the  measure- 
ment in  feet  is  no  less  accurate,  though  having 
only  one  decimal  place,  than  the  number  1.3420 
which  has  four  decimal  places. 

The  precision  of  a  measurement,  then,  is  in- 
dicated by  the  number  of  significant  figures. 
Measurement  is  actually  much  less  accurate 
than  is  generally  supposed.  A  carpenter's  care- 
ful measurements  are  usually  made  to  three  fig- 
ures, an  ordinary  surveyor's  to  four  figures,  a 
civil  engineer's  measurements  of  a  city  lot  to 
five  figures.  The  most  careful  measurements  pos- 
sible are  those  of  the  international  standards  of 
length,  like  the  "prototype  meters"  which  serve 
24 


SOME  RULES   OF  THUMB 

as  the  legal  standards  of  length  in  the  United 
States.  These  are  bars  made  of  a  durable  alloy, 
kept  in  a  sealed  room  at  a  constant  temperature. 
On  the  bars  the  length  of  a  meter  is  supposed 
to  be  indicated  by  five  scratches.  The  scratchef 
are  actually  from  6  to  8  microns  wide,  and  th< 
length  indicated  by  them  cannot  be  said  to  be 
really  ascertainable  within  .2  of  a  micron.  That 
is,  the  limit  of  human  accuracy  in  the  most  im- 
portant measurements  ever  made  is  seven  signi- 
ficant figures;  while  even  five-figure  measure- 
ment requires  expensive  instruments  and  trained 
skill. 

Bearing  these  things  in  mind,  one  may  easily 
see  waste  in  ordinary  computation.  This  is  no 
place  to  present  the  argument  for  "  contracted  " 
multiplication  and  division,  which  good  teachers 
of  mathematics  are  now  using  for  the  prevention 
of  this  waste.  The  greatest  obstacle  to  teaching 
these  common-sense  improvements  is  the  unwill- 
ingness of  elementary  teachers  to  have  their 
pupils  begin  learning  to  multiply  from  the  left 
of  the  multiplier  instead  of  from  the  right.  In 
the  sixteenth  century  both  methods  were  taught  ; 
25 


TEACHING  MATHEMATICS 

chance  seems  to  have  favored  the  right-hand 
end,  unluckily,  and  though  the  custom  now  turns 
out  to  be  foolish  it  is  intrenched  in  the  proverbial 
conservatism  of  school-teachers. 

Responsibility  and  self-reliance  are  promoted 
by  the  practice  of  "  checking  "  results  of  com- 
putation. The  well-known  check  by  "  casting 
out  nines"  is  valueless  except  where  all  the 
figures  used  are  kept  to  the  end.  Multiplication 
can  be  effectively  checked  by  interchanging 
multiplier  and  multiplicand;  division,  by  multi- 
plying the  quotient  by  the  divisor.  The  solu- 
tions of  algebraic  problems  should  always  be 
checked  by  substituting  the  answers  /';/  tJie 
words  of  the  problem  and  verifying  that  its  con- 
ditions are  fulfilled. 

The  square  root  of  a  number  is  needed  al- 
most from  the  beginning,  but  the  traditional 
method  for  finding  it,  besides  being  difficult  for 
the  pupil  to  understand  and  remember,  seems  to 
him  somewhat  artificial  also,  and  remote  from 
the  definition  of  a  square  as  the  product  of 
equal  factors.  If  it  were  not  for  the  necessity  of 
emphasizing  this  definition,  the  use  of  a  table 
26 


SOME  RULES   OF  THUMB 

of  square  roots  would  be  just  as  clearly  ad- 
visable as  the  use  of  a  table  of  sines  or  tangents 
is  in  trigonometry.  The  best  method  to  teach 
under  the  circumstances  is  the  so-called  "  guess 
and  try"  method,  although,  as  will  be  readily 
seen,  it  is  no  more  guess-work  than  the  method 
ordinarily  used. 

Thus,  let  it  be  required  to  find  the  square 
root  of  830.0.  Since  the  square  of  25  is  625  and 
the  square  of  30  is  900,  let  us  try  27.00  as 
the  square  root  required. 

27.00)830.0(30.74 
810.0 


Here  we  have  two  unequal  factors  of  830.0, 
namely,  27.00  and  30.74  ;  the  square  root  must 
be  between  them.  Try  the  number  half  way 
bettveen: 

27 


TEACHING  MATHEMATICS 

28.72)830.0(28.90 
574-4 
2556 
2298 

258 

258 


And  again: 


28.81)830.0(28.81 
5762 


^  This  method  is  based  on  the  ( erroneous  ) 
assumption  that  if  the  number  given  is  repre- 
sented by  d^,  and  if  our  guess  has  an  error  b,  then 
the  quotient  obtained  by  using  it  as  a  divisor  will 
have  the  same  error  b  in  the  opposite  direction  ; 
in  other  words,  that  the  factors  obtained  for  the 

*  The  argument  here  given  should  certainly  not  form 
a  part  of  the  first-year  work. 
28 


SOME  RULES  OF  THUMB 

given  number  would  be  a  +  b  and  a  —  b,  the 
quotient  and  the  divisor.  The  product  of  these 
two  factors,  however,  is  not  a^,  but  a^  —  b"^.  For 
example,  when  we  had  the  two  factors  a  —  b 
=  28.72  and  a-\-b  =  28.90  the  value  of  b  was  .09, 
and  32  =  .008 1,  which  would  not  affect  the  last 
figure  of  the  given  number.  If  we  had  realized 
that,  there  would  have  been  no  need  of  the  last 
division. 

It  would  be  convenient  if  we  could  be  sure 
that  when,  as  above,  the  value  of  b  comes  in  the 
last  place  (e.g.  within  the  fourth  figure  in 
a  four-figure  number)  the  true  value  of  the  root 
can  be  found  by  averaging  the  divisor  and 
quotient,  as  above.  This  is  always  the  case 
for  four-figure  numbers,  for  in  that  case  b  is  less 
than  I  in  the  third  figure  of  a  and  consequently 
b"^  is  less  than  i  in  the  fifth  figure  of  a^.  So 
much  the  more  is  it  true  for  a  greater  number  of 
figures.  Three-figure  numbers  present  little  dif- 
ficulty. 

This  method  possesses  the  advantage  that  it  is 
easily  remembered,  and  the  further  advantage 
that  it  is  a  constant  reminder  of  the  definition 
29 


TEACHING  MATHEMATICS 

of  square  root.  It  can  be  applied  to  cube  root, 
by  using  the  trial  root  as  a  divisor  twice  suc- 
cessively, and  then  averaging  the  second  quotient 
with  the  two  divisors.  There  is  a  greater  error, 
but  on  the  whole  the  process  is  not  more 
cumbrous  than  the  traditional  algorithm  for 
cube  root.  Cube  root,  however,  does  not  have  to 
be  used  much  in  high  school  mathematics,  and 
not  at  all  in  the  first  part ;  when  it  is  needed 
cube  root  tables  or  logarithms  would  ordinarily 
be  available. 

It  is  regrettable  that  college  examinations  still 
insist  on  the  square  root,  and  sometimes  on  the 
cube  root,  of  algebraic  polynomials.  The  require- 
ment is  not,  however,  so  significant  as  it  might 
seem,  for  the  following  reasons : 

First,  because  no  self-respecting  examiner  can 
give  an  example  in  algebraic  square  or  cube  root 
that  does  not  "come  out  even"  ;  unless  he  speci- 
fies restrictions  (on  the  relative  size  of  the  num- 
bers represented  by  the  letters)  that  are  out  of 
the  question  in  elementary  algebra. 

Second,  because  such  examples,  if  they  do 
come  out  even,  can  easily  be  solved  by  inspec- 
30 


SOME  RULES  OF  THUMB 

tion,  as  far  as  those  that  have  four  terms  in  the 
root.  There  is  a  little  difficulty  about  the  signs 
when  there  are  four  terms,  but  nothing  that  a 
pupil  will  not  learn  to  do  with  glee,  to  avoid 
remembering  either  of  the  rules  that  vexed  our 
younger  days. 

Third,  because  neither  of  the  rules  is  of  any 
use  except  for  emphasizing  two  simple  cases  of 
the  binomial  theorem  ;  and  if  the  schools  stop 
teaching  them  with  the  motive  of  saving  the 
pupil's  effort  for  things  more  worth  while,  the 
colleges  will  stop  examining  for  them. 

Whatever  is  done  with  the  "  evolution "  of 
polynomials  should  be  reserved  for  the  mora 
formal  discussions  of  the  second  or  third  year. 


V 

GEOMETRY   AS  ALGEBRAIC   MATERIAL 

The  formulae  for  the  measurement  of  triangles 
and  quadrilaterals,  the  principal  facts  about  sim- 
ilar figures,  the  Pythagorean  theorem,  and  the  use 
of  similar  right  triangles  for  the  indirect  meas- 
urement of  distances  are  all  of  great  value  in  fur- 
nishing material  for  the  algebraic  work  of  the 
first  year.  They  not  only  illustrate  the  rational 
derivation  and  the  numerical  application  of  for- 
mulae, but  they  serve  for  the  construction  of  equa- 
tions many  of  which  might  actually  occur  in 
practical  application,  and  all  of  which  strengthen 
the  mental  association  between  numbers  on  the 
one  hand  and  lengths  and  areas  on  the  other. 

The  high  school  pupil  begins  his  work  with  at 
least  one  mathematical  item  among  his  assets  : 
he  knows  that  the  area  of  a  rectangle  is  obtained 
by  multiplying  its  length  and  its  breadth.  All 
the  geometrical  theorems  referred  to  above  can 
be  founded  upon  that ;  without  too  great  insist- 
32 


GEOMETRY  AS  ALGEBRAIC  MATERIAL 

ence  on  the  proof  of  reasonably  obvious  things, 
ji  good  idea  can  be  given  of  successive  logical 
dependence ;  and  for  the  rectangle-rule  itself  a 
proof  can  be  given  that  on  the  one  hand  relates 
to  his  previous  knowledge,  and  on  the  other  hand 
prepares  him  for  a  rigid  "  limit  "  proof  in  his 
later  work. 

This  program  makes  necessary  a  considerable 
departure  from  the  order  of  topics  and  from  the 
methods  of  presentation  that  seem  appropriate 
for  geometry  taught  by  itself  in  the  second  stage 
of  high  school  mathematics.  Only  a  sketch  can 
be  given  here,  but  one  innovation  in  terminology 
may  be  mentioned.  This  is  the  word  "stripe," 
used  in  German  school-books  for  the  figure  formed 
by  two  parallel  lines.  It  gives  a  name  to  a  con- 
figuration that  every  geometry  student  learns  to 
imagine,  for  example  when  he  thinks  of  two  par- 
allelograms or  triangles  with  the  same  altitude ; 
and  of  course  it  is  known  in  popular  use  with  its 
exact  (or  shall  we  say  scientific  .-')  meaning. 

Again,  a  "strip"  is  a  limited  portion  of  a 
stripe  ;  for  example,  an  inch-wide  strip  along  the 
bottom  of  a  triangle. 

33 


TEACHING   MATHEMATICS 

The  proof  for  the  area  of  a  rectangle  is  briefly 
summarized  as  follows.  The  teacher  will  readily 
adapt  it  to  the  more  concrete  point  of  view  that 
the  pupil  regards  it  from. 

If  the  length  of  a  rectangle  can  be  expressed 
by  a  decimal  number,  the  area  of  the  unit-wide 
strip  along  the  side  of  the  rectangle  can  be  ex- 
pressed by  the  same  number.  For  on  each  unit 
of  length  stands  the  unit  square,  which  is  the 
unit  of  area ;  on  each  tenth  of  a  unit  stands  a 
portion  of  the  unit  square  congruent  with  every 
other  such  portion,  that  is,  a  tenth  of  the  unit 
area  ;  and  so  on. 

If  also  the  width  of  the  rectangle  can  be  ex- 
pressed by  a  decimal  number,  the  area  of  the 
rectangle  can  be  expressed  by  the  product  of  that 
number  by  the  length-number.  For  each  unit  in 
the  width  is  one  end  of  a  unit-wide  strip  whose 
area-number  is  the  length-number  ;  each  tenth 
of  a  unit  is  one  end  of  a  tenth  of  the  unit-wide 
strip  ;  and  so  on. 

Rhomboids,  triangles  and  trapezoids  can  then 
be  treated  as  usual.  Similar  triangles  are  covered 
by  the  following  succession  of  theorems.  [A  fig- 
34 


GEOMETRY  AS  ALGEBRAIC  MATERIAL 

ure  is  said  to  be  inscribed  in  a  stripe  when  its 
vertices  lie  in  the  sides  of  the  stripe.] 

Two  triangles  inscribed  in  the  same  stripe  have 
the  same  ratio  as  their  bases. 

If  two  triangles  have  one  angle  the  same,  their 
ratio  is  equal  to  the  ratio  of  the  products  of  the 
sides  that  include  the  equal  angles. 

If  two  triangles  have  the  angles  of  one  respect- 
ively equal  to  the  angles  of  the  other,  the  ratio 
of  any  two  corresponding  sides  is  equal  to  the 
ratio  of  any  other  two  corresponding  sides  ;  and 
the  ratio  of  their  areas  is  the  square  of  the  ratio 
of  two  corresponding  sides. 

In  all  these  proofs  the  cumbrous  algebra  of 
the  geometry  text-book  is  replaced  by  a  notation 
which  always  uses  a  single  letter 
for  one  number,  and  which  not 
only  represents  by  that  letter 
the  measurement-number,  but 
also  uses  it  for  the  name  of  the 
line,  or  the  angle,  or  the  area 
measured.  Thus,  in  the  accom-  fig.  i 

panying  figure,  vS  represents  the  area  of  the  upper 
triangle  S,  X  represents  the  area  of  the  quad- 
35 


TEACHING  MATHEMATICS 

rilateral  below  S,  and  vS  +  A"  is  the  name  of  the 
triangle  composed  of  the  quadrilateral  X  and  the 
triangle  6'. 

Angles,  and  the  number  of  degrees  in  an  angle 
or  in  an  arc,  are  represented  by  capital  letters  ; 
areas  are  represented  by  capital  letters ;  lines 
are  named  by  small  letters,  and  length-numbers 
are  represented  by  small  letters  (lower-case  let- 
ters). The  context  always  prevents  confusion  of 
signification,  and  the  great  advantage  is  gained 
of  similarity  of  notation  in  geometry  and  algebra. 

It  is  only  recently  that  text-books  of  geometry 
have  ventured  to  express  even  purely  algebraic 
proofs  in  the  ordinary  notation  of  algebra.  Equa- 
tions like  the  following  are  still  found  even  in 
the  best  of  recent  text-books  : 

aABC^AB    AC 
KDEF~  DE^  DF 

This  is  objectionable  not  only  because  the 
hasty  pupil  maybe  tempted  to  write  his  product 

as  -  ^       ,  but  also  because  the  attempt  is  made 

to  create  the  erroneous  impression  that  — J^lr 

36 


GEOMETRY  AS  ALGEBRAIC  MATERIAL 

is  a  ratio  between  the  geometrical  figures,  and  is 
some  mysterious  thing  different  from  the  ratio 
of  their  area-numbers. 

This  is  the  place  to  comment  on  the  vicious 
custom  of  lettering  angles  or  lines  with  Arabic 
numerals.  Imagine  the  confusion  in  stating  val- 
ues, e.g., 

the  line  12  =  13  ( !) 

Of  course  a  bright  pupil  can  be  made  to  skate 
over  these  places,  but  why  introduce  unneces- 
sary elements  of  confusion  when  there  are  plenty 
of  difficulties  with  the  clearest  notation  ?  The 
practice  of  draughtsmen  should  not  mislead  us  ; 
they  are  not  teaching ;  and  when  our  pupils  get 
to  the  draughting-table  they  can  learn  this 
custom  if  they  care  to  be  so  injudicious. 

The  single-letter  notation  should  be  adhered 
to,  if  for  no  other  reason,  because  all  the  equa- 
tions in  geometry  are  really  equations  between 
numbers  or  combinations  of  numbers.  There  is, 
however,  the  additional  reason  that  by  its  use  it 
is  easier  to  indicate  the  relations  of  duality  that 
appear  here  and  there  in  elementary  geometry  ." 
not  so  many  in  plane  as  in  solid. 
37 

94259 


TEACHING  MATHEMATICS 

These  considerations  will  be  my  excuse  for 
introducing  here  one  of  the  many  modifications 
of  Euclid's  proof  that  the  square  on  the  hypote- 
nuse of  a  right  triangle  is  the  sum  of  the  squares 
on  the  other  two  sides. 

The  proof  of  the  Pythagorean  theorem  can  be 
based  on  similar  triangles  or  on  the  rule  for  the 
area  of  a  trapezoid.  The  proof  given  below  is 
selected  because  it  brings  into  view  the  advan- 
tages of  a  good 
notation,  and 
at  the  same 
time  illustrates 
the  omission 
of  essential 
steps  in  a  proof 
for  the  sake  of  a 
unified  impres- 
sion on  the  be- 
ginner's mind. 
It  will  be 
noticed  that 
proofs  by  sup- 
erposition are  passed  over  as  obvious  at  this  stage. 
38 


B 

/      / 

^^•^ 

^^-^^ 

^>. 

C 

6 
Fig.  2 


GEOMETRY  AS  ALGEBRAIC   MATERIAL 

Here  the  square  c^  is  inscribed  in  the  same 
stripe  as  a  rhomboid  which  we  can  call  R  ;  and 
they  have  the  same  base  a.  Consequently 
Q)  R  =  a^ 
Again  the  rectangle  cr  is  inscribed  in  the  same 
stripe  as  a  rhomboid  which  we  can  call  S ;  and 
they  have  the  same  base  c.  Consequently 
(2^   S  =  ex 

A  tracing  of  the  rhomboid  R  can  be  made  to 
fit  exactly  on  the  rhomboid  S,  by  keeping  the 
corner  B  fixed  and  turning  the  tracing  of  R 
upon  that  as  a  pivot.  That  is, 

0^  =  5 

Substituting  equations  (T^  and  (2)  in  (j^  we  have 

r^S  a^  =  ex 
In  the  same  way,  on  the  other  side  of  the  tri- 
angle, we  obtain 

and  by  adding  these 

^6^  cP'  -^^  b"^  =  ex  -V  cy 
or  0  ^2  +  ^2  ==  ^2 
39 


TEACHING  MATHEMATICS 

It  is  common  enough  to  illustrate  the  applica- 
tion of  similarity  by  estimating  the  height  of  an 
object  by  the  length  of  its  shadow.  Some  recent 
books  even  introduce  the  sine,  cosine,  and  tangent 
into  plane  geometry.  Certainly  it  seems  wise  to 
make  use  of  measurable  angles,  and  to  show  that 
the  measurement-numbers  of  inaccessible  lines 
can  be  computed. 

For  the  first-year  work,  however,  a  clearer  im- 
pression will  be  made  if  we  confine  ourselves  to 
one  of  the  six  ratios  of  the  sides  of  the  right  tri- 
angle. The  most  immediately  applicable  is  the 
tangent.  A  three-figure  table  of  tangents  can  go 
on  a  single  page,  without  the  confusion  attend- 
ant upon  reading  backwards  for  angles  over  45°. 
Solving  triangles  in  which  angles  are  included 
among  the  data  will  then  give  practice  in  appli- 
cations of  algebra  and  geometry  which  have  an 
aspect  of  reality,  not  only  because  they  are  ob- 
viously such  as  human  beings  have  to  work  at, 
but  also  because  the  arithmetic,  though  compli- 
cated, does  not  seem  to  have  been  purposel;' 
complicated  for  the  good  of  the  pupil. 


40 


VI 

THE  GRAPHICAL  METHOD 

The  graphical  method  of  comparing  numerical 
data  and  of  exhibiting  statistics  is  of  wide  pop- 
ular use.  We  compare  crudely  the  numerical 
strength  of  armies  and  navies  by  the  pictures  of 
giants  and  dwarfs  in  the  uniforms  appropriate  to 
the  service,  or  by  sketches  of  battleships  white 
for  our  side  and  black  for  the  other.  Census  re- 
ports, magazine  articles,  and  even  schoolboy  com- 
positions compare  the  cotton  crop  or  the  popu- 
lation of  different  states  by  black  lines  in  length 
proportional  to  the  numbers.  Beet-sugar  and 
cane-sugar  production  are  exhibited  in  curves  that 
sweep  upward  and  cross  at  a  significant  date.  The 
political  muck-raker  sets  out  the  expense  accounts 
that  he  is  hunting  in  a  curve,  and  delights  to 
locate  his  adversary's  extravagance  by  its  undu- 
lation. It  has  become,  in  the  course  of  the  last 
generation,  a  mode  of  public  expression  ;  it  is 
addressed  to  the  intelligent ;  it  should  be  a  part, 
41 


TEACHING  MATHEMATICS 

therefore,  of  the  education  of  youth  ;  and  its  place 
is  in  the  mathematics  course. 

Probably  the  first  point  at  which  the  advan- 
tage of  this  method  is  obvious  to  the  pupil  is  in 
the  discussion  of  elimination  between  two-letter 
equations.  For  example,  the  equation 

5^-37=20 
has  a  list  of  number-pairs  that  will  satisfy  it,  in 
other  words,  "answers";  and  there  is  no  limit  to 
the  number  of  such  answers.  Yet  there  are  num- 
ber-pairs that  will  not  satisfy  this  equation.  The 
equation  implies  a  restriction  upon  number-pairs. 
What  sort  of  restriction  .? 

We  begin  by  defining  an  algebraic  scale  as  a 
straight  line  upon  which  one  point  is  marked  for 
zero,  and  every  other  point  represents  some  num- 
ber, positive  or  negative,  the  integers  succeeding 
each  other  at  convenient  equal  intervals.  Two  of 
these  scales,  set  perpendicular  to  each  other  with 
the  zero-points  together,  we  call  axes.  Every 
point  in  the  plane  is  then  opposite  to  two  num- 
ber-points in  the  axes,  one  in  each,  and  repre- 
sents that  number-pair.  It  is  a  highly  artificial 
device,  but  so  is  algebra;  and,  in  fact,  arithmetic. 
42 


THE  GRAPHICAL  METHOD 

Now  the  points  that  represent  the  number* 
pairs  serving  as  answers  to  the  equation  we 
started  with  all  lie  on  a  straight  line.  The  pupil 
will  infer  that  himself,  when  he  has  plotted  a 
few.  The  proof  that  all  the  points  satisfying  (i.  e., 
representing  number-pairs  that  satisfy )  a  two- 
letter  equation  of  the  first  degree  lie  on  a  straight 
line  is  one  of  the  simplest  "  exercises  "  in  similar 
triangles  ;  instead  of  being  reserved  for  college,  it 
is  well  within  first-year  high  school  work.  It  is 
probably  the  easiest  and  clearest  example  of  a 
locus,  much  clearer  than  the  bisector  proposi- 
tions, which  foster,  temporarily  at  least,  or  among 
the  laggards,  the  notion  that  a  "  locus-of-points  " 
is  some  kind  of  a  bisecting  line.  It  is  also  a 
valuable  argument  to  refer  to  later,  when  the 
time  comes  to  speak  of  "  necessary  and  sufficient 
conditions,"  or  propositions  and  their  converses. 

This  representation  of  equations  is  not  merely 
for  illustration,  but  is  also  a  means  of  checking 
the  solution  of  a  pair  of  two-letter  equations. 
The  pupil  can  easily  estimate  the  intercepts  of 
each  equation  on  the  axes,  and  thus  locate 
the  two  lines,  and  their  intersection,  which  sat- 
43 


TEACHING  MATHEMATICS 

isfies  both  equations.  He  will  balk  at  this  if 
he  is  expected  to  draw  the  lines  for  every  one 
of  a  dozen  number-pairs  ;  a  much  easier  plan  is 
to  rule  a  line  on  a  sheet  of  tracing  paper,  lay 
that  line  through  one  pair  of  intercepts  and  a 
straightedge  through  the  other.  The  point  thus 
located  will  roughly  check  his  answer. 

The  objection  will  be  raised  that  this  is  teach- 
ing analytic  geometry  in  the  high  school ;  those 
to  whom  that  objection  seems  reasonable  can- 
not be  argued  with.  Others  will  see  that  the 
method  furnishes  a  good  point  of  contact 
between  the  pupil's  algebra  and  his  geometry, 
and  increases  his  command  over  both.  Even 
more  so  the  next  suggestion,  perhaps  for  second- 
year  work — not  later. 

Why  should  not  the  pupil  be  taught  that 
every  circle  has  an  equation  in  the  form 

x^  +JJ/2  -\- ax  ^^  by -\- c  ^^  o 
and  conversely  that  one  and  only  one  circle  can 
be  constructed  for  every  such  equation }  No 
knowledge  of  geometry  beyond  the  Pythagorean 
theorem  is  required,  and  a  great  addition  is  made 
to  the  pupil's  capacity. 

44 


THE  GRAPHICAL  METHOD 

Be  it  remembered  that  these  two  things  —  the 
circle-equation  as  well  as  the  straight-line  equa- 
tion—  are  proposed  as  additions  to  the  work  of 
the  first  two  years.  There  is  one  advantage  to 
this  which  every  teacher  of  experience  will  ap- 
preciate ;  that  is,  that  "  real  problems  "  in  elimin- 
ation, which  have  been  scarce,  are  at  hand  here 
in  plenty.   For  example:  — 

Find  the  point  of  intersection  of  AB  and  PQ, 
where  the  number-pairs  of  the  four  points  are 
respectively  as  follows  :  A,  5,6  ;  B,   1,2  ;  P,  5,1  ; 

Q,  2,7. 

We  can  take  for  AB  the  equation  ax+by=  i, 
and  for  /"(J  the  equation /;ir+^j=i ;  substituting 
the  values  for  A,  that  is,  x=^  andjj/  =  6,  we  get 
5  «  +  6^=i  ;  2indior  B,a-\-2  b-=i,  whence  a  =  —  i 
and  b=i  ]  in  the  same  way  5 /  +  ^  =  i  and  2p  +  yq 
=  1,  whence  p  =  -^^  and  q  =  -^^.  Then  for  AB 
the  equation  is— ;f+ J  =  i  and  for  P^,  2x+y=\i  ; 
whence  x=T)^  and  j  =  4,  as  the  number-pair  lo- 
cating the  required  intersection.  This  answer  can 
of  course  be  readily  checked  by  the  tracing-paper 
and  straightedge,  as  before. 

The  circle  through  three  points  would  in 
45 


TEACHING  MATHEMATICS 

the  same  way  give  three-letter  equations  of  the 
first  degree  to  determine  the  coefficients  in  the 
equation 

thus  for  the  circle  through  the  three  points 
x  =  2,  y  =  3;  x=4,  jj/=5  ;  x  =  3,  y=i;  we  have 
the  three  equations 

2^  +  3  b  +  c=  —  13 

3  a  +  b  +  c=  — 10 

There  are  other  exercises  which  will  at  the 
same  time  throw  side  lights  on  elimination  and  on 
the  fundamental  theorems  of  geometry.  These 
will  readily  suggest  themselves  to  an  enterpris- 
ing teacher. 

A  very  important  exercise  is  the  careful  plot- 
ting of  a  curve  of  squares.  The  squares  of  the 
numbers  from  i  to  10  being  plotted  in  class, 
each  section  between  successive  integers  can  be 
assigned  to  a  small  group  of  pupils  to  locate  the 
intermediate  points  for  tenths  [(2.1)^,  (2.2)^,  (2.3)^, 
and  so  on] ;  this  will  serve  to  illustrate  the  con- 
venience of  arithmetical  interpolation  in  the  flat 
regions  of  the  curve. 

46 


THE  GRAPHICAL  METHOD 

It  is  very  instructive  to  plot  the  areas  of  fig- 
ures that  vary  under  some  restriction  (say  squares, 
similar  rectangles,  similar  triangles,  circles,  par- 
allelograms with  the  sides  given  but  not  the  an- 
gles, and  so  on),  so  that  one  number  will  serve 
to  determine  the  area  of  one  figure  in  the  series. 
The  mere  fact  that  an  area-number  is  repre- 
sented by  a  line  is  significant. 

The  use  of  a  "  standard  parabola  "  (curve  for 
y=x^)  has  been  suggested  for  checking  the 
solution  of  one-letter  quadratics.  Thus  for  the 
quadratic 

we  take  a  carefully  plotted  curve  ior  y=x'^,  find 
the  intercepts  on  the  same  axes  for 

y=-px-q 
and  lay  a  straightedge  through  them.    The  in- 
tersections of  this  line  with  the  curve  ior  y^x^ 
will  have  x^^  —px—q,  that  is, 

x'^-\-px+q=o 
and  the  values  of  x  will  be  the  roots  of  the  equa- 
tion. 

It  is  not  advisable  to  go  into  the  discussion  of 
ellipses,  parabolas,  or  hyperbolas  in  high  school 
47 


TEACHING  MATHEMATICS 

work,  or  to  have  the  pupil  expected  to  plot  every 
two-letter  quadratic  that  he  deals  with.  Many  of 
the  curves  that  he  would  have  to  deal  with,  es- 
pecially very  broad  or  very  narrow  hyperbolas, 
would  be  difficult  for  him  to  recognize  from  the 
few  roughly-plotted  points  he  can  get ;  and  when 
we  realize  that  plotting  an  algebraic  curve  by 
points  is  a  stupid  and  profitless  job  anyway,  one 
that  a  college  student  will  be  taught  to  dodge 
wherever  possible,  it  is  just  as  well  for  high 
schools  to  confine  themselves,  for  the  most  part, 
to  simple  curves  that  can  be  utilized  for  collat- 
eral information  ;  for  help  on  the  algebra  or  on 
the  geometry  that  forms  the  main  part  of  his 
present  burden. 

Nor,  on  the  whole,  is  it  advisable  to  resort  to 
the  plotting  of  parabolas  often  used  to  "explain" 
the  double  or  the  imaginary  roots  of  a  one-letter 
quadratic.  These  can  be  exhibited  much  more 
easily  and  simply  by  the  straightedge  and  stand- 
ard parabola,  as  above. 


VII 

THE  BASES  OF   PROOF  IN   GEOMETRY 

Euclid  (1,4)  uses  the  following  phraseology:  "If 
we  fit  the  triangle  ABF  upon  the  triangle  AEZ, 
and  if  we  put  not  only  the  point  A  upon  the  point 
A,  but  also  the  straight  line  AB  upon  the  straight 
line  AE  ...*';  so  much  of  motion,  then,  has 
always  been  regarded  as  orthodox,  though  some 
recent  purists  object  even  to  that.  The  motions 
implied  by  Euclid  in  this  quotation  constitute  no 
insignificant  part  of  the  definition  of  a  plane ;  it 
is  implied  also  that  a  geometrical  figure  can  be 
moved  about  without  distortion.  Again,  it  is  im- 
plied, though  Euclid  does  not  mention  it  or  use 
the  inference  otherwise,  that  a  plane  figure  can 
be  overturned  upon  its  plane,  so  that  the  order 
of  the  parts  of  one  of  the  triangles  may  be  re- 
versed; for  he  makes  no  exception  to  his  theo- 
rem. 

Although  a  first  treatment  of  geometry  should 
49 


TEACHING  MATHEMATICS 

not  include  proofs  of  theorems  that  do  not  cry  for 
proofs,  and  although  it  is  quite  out  of  the  ques- 
tion to  point  out  all  the  assumptions  that  are 
logically  necessary  and  no  others,  it  is  desirable 
to  take  up  at  some  time  in  the  high  school 
course  a  system  of  propositions  based  upon 
assumptions  that  will  seem  to  the  pupil  of  that 
age  necessary  and  sufficient,  so  that  he  can  ar- 
rive at  an  appreciation  of  the  "monument  of 
human  reason  "  that  systematic  elementary  geo- 
metry has  for  centuries  been  to  the  world.  He 
should  have  had  a  previous  experience  in  geo- 
metry demonstrations,  so  as  to  have  an  appetite 
for  a  cogent  argument ;  he  should  have  had 
an  experience  in  the  perception  of  successive 
logical  dependence,  so  as  to  have  a  general  no- 
tion of  what  he  is  driving  at  when  he  starts  in. 

As,  in  algebra,  we  cannot  justify  to  the 
beginner  the  use  of  an  equation  as  a  means 
of  simplifying  his  work,  so  in  systematic  geome- 
try we  cannot  justify  to  the  beginner  the  early 
proofs  as  necessary  to  his  acceptance  of  their 
conclusions.  At  least  one  of  our  basic  as- 
sumptions—  the  axiom  of  parallels  —  is  in  mod- 
50 


THE  BASES   OF  PROOF  IN  GEOMETRY 

ern  times  known  to  be  only  an  inference 
from  experience.  Many  of  the  propositions 
that  we  carefully  prove  can  also  be  so  regarded. 
The  only  justification  for  the  proofs  is,  that  we 
want  to  reduce  to  as  small  a  compass  as  we 
conveniently  can  the  facts  for  which  we  must 
depend  upon  observation,  so  that  experience 
can  be  foretold.  In  this  way  a  comparatively 
small  number  of  elementary  truths,  with  standard 
methods  of  inference,  will  serve  to  connect 
and  unify  a  large  number  of  facts  that  might 
to  hasty  observation  seem  independent. 

For  this  reason  the  first  group  of  theorems 
should  be  confined  to  simple  figures,  and  should 
serve  to  illustrate  standard  methods  of  inference  ; 
first,  by  utilizing  the  characteristic  qualities  of  a 
plane  surface,  and  next,  by  obtaining  numbers 
called  measurement-numbers,  by  which  a  geo- 
metric magnitude  can  be  reconstructed  out  of 
standard  or  unit  magnitudes  of  the  same  kind. 

The  characteristic  qualities  of  a  plane  can  be 
appreciated  by  comparing  it  with  other  surfaces 
which  do  not  possess  all  of  these  qualities. 
They  are :  — 

51 


TEACHING  MATHEMATICS 

I.  Any  figure  in  a  plane  can  be  moved  about 
upon  it  until  any  specified  point  of  the 
figure  coincides  with  any  specified  point 
of  the  plane.  This  motion  will  be  called 
sliding. 
II.  Any  figure  in  a  plane  can  be  rotated 
about  any  specified  point  of  the  figure, 
until  any  specified  line  of  the  figure 
through  that  point  coincides  with  any 
specified  line  of  the  plane  through  that 
point.  This  motion  will  be  called  rotation. 
III.  Any  figure  in  a  plane  can  be  taken  out 
of  the  plane,  turned  over,  and  laid  back 
upon  the  plane,  so  that  the  order  of  its 
parts  is  thereby  reversed.  This  motion 
is  called  overturning. 

For  all  of  these  motions  the  figure  must  be 
thought  of  as  a  separate  thing  from  the  plane, 
though  the  figure  lies  in  it,  and  may  be  named 
as  if  a  part  of  it.  Tracing  paper  furnishes  a  con- 
venient means  of  illustration. 

On  a  cylindrical  surface  (cylinder  of  revolu- 
tion) the  first  motion,  sliding,  is  the  only  one  of 
the  three  that  is  possible  without  distortion  ;  on 
a  spherical  surface  the  first  two  are  possible,  but 
not  the  third ;  on  the  plane  only  are  the  three 
motions  possible, 

52 


THE  BASES   OF  PROOF  IN  GEOMETRY 

The  proof,  therefore,  that  circles  are  congruent 
if  they  have  equal  radii  is  valid  for  a  cylinder  (if 
one  can  succeed  in  defining  what  a  circle  is  on  a 
cylinder) ;  but  the  proof  that  the  diameter  of  a  cir- 
cle divides  it  into  two  congruent  parts  is  not  valid 
for  a  cylinder,  though  it  is,  if  properly  chosen,  valid 
fora  sphere.  Symmetry  propositions,  being  proved 
on  a  plane  by  overturning,  cannot  be  regarded 
as  true  for  a  sphere  without  additional  argument. 

The  question  of  order  of  parts  requires  care- 
ful teaching.  The  pupil  must  be  reminded  that 
the  figure  L  cannot  be  got  into  the  position  J 
by  any  amount  of  sliding  and  rotating.  He  must 
be  shown  that  there  is  not  only  a  right  and  left 
order,  but  a  clockwise  and  a  counter-clockwise 
order ;  and  should  be  taught  to  name  the  parts 
of  a  figure  with  attention  to  this  distinction. 
Tracing  paper  again  is  a  ready  means  of  experi- 
ment for  this  purpose ;  it  is  well  to  letter  the 
first  diagrams  with  such  letters  as  T,  U,  V,  to 
confine  the  pupil's  attention  to  the  figures  them- 
selves, and  not  bother  him  with  the  strange  ap- 
pearance of  reversed  unsymmetrical  letters,  like 
B,  F,  G,  and  so  on. 

53 


TEACIHNG  MATHEMATICS 

All  this  matter  of  reversal  is  within  the 
experience  of  every  one  who  has  seen  for  example 
the  lettering  on  a  glass  door;  nevertheless  it  will 
be  interesting  to  the  pupil  as  a  scientific  point  of 
view  for  his  vague  experience.  And  though  the 
proof  of  congruence  propositions  such  as  the  fol- 
lowing will  be  made  more  detailed  than  under 
the  loose  treatment  hitherto  customary,  the  ar- 
gument will  be  more  interesting  to  him,  and 
more  satisfactory. 

Let  us  take   the   proposition  Euclid  I,  4,  to 
which  I  have  previously  referred  : — 

If  two  triangles  have  two  sides  and  the  in- 
cluded angle  of  one  equal 
respectively  to  the  corres- 
ponding parts  of  the  other, 
the  triangles  are  congruent. 
Suppose  a=x,  b=y,  and 
the  angle  Z=  C,  all  in  the 
same  order  in  the  two  fig- 
ures. 

To  prove   ABC=XYZ. 
Slide  ^FZ'untilthe  point 
Z  comes  to  C.  Rotate  XVZ  about  6"  as  a  pivot 
54 


THE   BASES   OF  PROOF  IN  GEOMETRY 

until  y  fits  exactly  on  b.  Then  the  point  X  will 
fit  exactly  over  A,  because  j/  =  <^. 

Since  the  parts  are  in  the  same  order,  Cand 
Z  will  be  on  the  same  side  of  b,  and  x  will  fit  on 
a  because  the  angle  C=Z\  and  the  point  Fwill 
then  fit  exactly  over  B  because  x=a. 

Then  all  the  vertices  of  XYZ  exactly  cover 
the  vertices  of  ABC,  and  the  triangles  are  con- 
gruent. 

Again,  suppose  a=x,  b=y,  and  the  angle  C=Z, 
all  the  parts  in  ABC  being  in  reverse  order  to 
XYZ. 

To  prove,  as  before,  ABC ^ XYZ. 

Overturn  XYZ,  so   that 
the  order  of  parts  will  be 
the  same  as  ABC.    Then 
proceed  with  the  same  -^' 
proof  as  before. 

The  motion  of  sliding 
could  be  exhibited  as  a  pure 
translation  by  drawing  B  Y, 
to  cut  ZX  as  at  Q,  and  Z' 
then  sliding  XYZ  along 
B  Y  until  Y  reaches  B ;  then  we  could  rotate 
55 


TEACHING  MATHEMATICS 

about  B  until  ;r  falls  on  a  and  so  on.  The  objec- 
tion is  of  course  the  artificial  character  of  the 
device  ;  and  for  the  sake  of  a  theoretical  ad- 
vantage, remote  at  best,  it  seems  hardly  worth 
while. 

Figures  are  symmetrical  about  an  axis  when 
one  can  be  made  to  fit  exactly  upon  the  other  by 
overturning  its  half  of  the  plane  about  that  line, 
as  a  door  upon  its  hinge.  By  analogy  with  a  mir- 
ror, one  of  the  figures  may  be  called  the  image 
of  the  other. 

Two  symmetrical  lines  intersect  the  axis  at  the 
same  point  and  make  equal  angles  with  it ;  two 
symmetrical  points  are  in  a  straight  line  per- 
pendicular to  the  axis,  and  are  equally  distant 
from  it. 

A  very  important  theorem,  and  one  that  should 
be  placed  early,  is  that  the  figure  formed  by  two 
intersecting  circles  is  symmetrical  about  its  line 
of  centres.  This  theorem  enables  us  to  prove 
immediately  the  congruence  of  mutually  equilat- 
eral triangles. 

When  we  have  proved  the  three  propositions 
about  congruent  triangles,  and  the  two  converse 
56 


THE  BASES   OF  PROOF  IN  GEOMETRY 

propositions  about  the  angles  of  a  stripe,^  the 
whole  subject  of  congruence  and  symmetry  in  a 
plane,  and  the  whole  subject  of  mensuration  for 
plane  rectilinear  figures  is  within  our  reach. 

In  selecting  an  order  of  theorems  the  teacher 
should  avoid  successive  dependence  where  it  is 
not  necessary ;  for  example,  neither  of  the  two 
propositions  about  the  angles  of  a  stripe  should 
be  made  to  depend  upon  the  other.  Neglect  of 
this  precaution  will  lead  to  distrust  of  perfectly 
good  proofs. 

The  theorems  about  the  angles  at  the  base  of 
an  isosceles  triangle,  and  the  converse  theorem, 
can  be  proved  by  overturning  the  triangle  and 
showing  it  to  be  congruent  with  its  old  position. 

The  theorem  that  only  one  perpendicular  can 
be  drawn  from  a  point  to  a  straight  line  has  an 
amusing  proof.  Overturn  the  figure  about  one  of 
the  two  lines  ;  if  both  were  perpendicular  this 
could  be  repeated  indefinitely,  the  foot  of  each 
falhng  in  the  straight  line,  until  the  straight  line 
returned  into  itself.  To  tell  the  truth,  I  have  never 

1  If  the  two  lines  are  parallel, certain  angles  are  equal; 
if  certain  angles  are  equal,  the  two  lines  are  parallel. 

57 


TEACHING  MATHEMATICS 

found  a  pupil  who  considered  this  as  amusing  as 
it  seemed  to  me. 

The  congruence  of  figures  cannot  be  proved 
without  overturning  except  when  the  order  of 
parts  is  immaterial ;  that  is,  when  the  figures  to 
be  proved  congruent  are  themselves  symmet- 
rical. 

Among  the  very  first  theorems  then,  for  sim- 
plicity of  proof  are  :  — 

If  two  central  angles  are  equal  their  arcs  are. 

If  two  arcs  are  equal,  their  central  angles  are. 

If  two  arcs  are  equal,  their  chords  are. 

If  two  chords  are  equal,  their  arcs  are. 

The  first  two  of  these  are  at  the  foundation  of 
measurement;  their  position  at  the  beginning  of 
the  subject  is  therefore  a  strategic  advantage. 

The  execution  of  problems  of  construction  in 
geometry  has  no  logical  connection  with  the  de- 
velopment of  theorems,  except  where  the  con- 
structions show  the  existence  of  the  figures  re- 
ferred to  in  the  theorems.  The  question  of  the 
existence  of  such  figures  is  settled  by  showing 
that  their  properties  are  not  contradictory,  either 
of  each  other,  or  of  the  general  postulates  of  space 
58 


THE  BASES  OF  PROOF  IN  GEOMETRY 

in  which  they  are  assumed  to  exist.  In  most 
cases,  therefore,  since  this  consistency  is  shown 
by  the  extensive  development  of  systems  of  the- 
orems based  upon  the  properties  of  the  figures 
in  question,  their  existence  may  safely  be  as- 
sumed and  this  particular  requirement  for  the 
execution  of  constructions  may  safely  be  ignored. 

On  the  other  hand,  they  do  furnish  excellent 
practice  in  the  application  of  theorems,  and 
should  be  freely  used  for  this  purpose.  When, 
however,  the  methods  of  construction,  instead 
of  conforming  to  the  practice  of  draughtsmen, 
confine  themselves  to  the  two  Euclidean  instru- 
ments, —  compass  and  unmarked  straightedge, 
—  the  historical  and  the  logical  reasons  for  that 
limitation  should  be  pointed  out. 

None  of  the  inequality  theorems  has  any  nec- 
essary application  in  the  early  part  of  geometry, 
though  they  are  generally  placed  early.  They 
should  follow  the  measurement  theorems  rather 
than  precede  them.  They  are  necessary  to  the 
theorems  on  the  value  of  tt,  and  to  those  only. 


VIII 

THE  METHOD  OF  LIMITS 

The  direct  measurement  of  a  quantity  is  ac- 
complished by  dividing  it  up  into  parts  that  are 
congruent  with  the  unit  or  with  submultiples  of 
the  unit.  Now  there  are  some  quantities  which 
cannot  be  thus  decomposed.  The  diagonal  of  a 
square  whose  side  is  the  unit  of  length  is  such  a 
quantity.  Its  length,  in  terms  of  the  unit,  can- 
not be  exactly  expressed  in  figures,  for  if  it 
could,  these  figures  would  represent  parts  of  the 
line  that  would  be  multiples  either  of  the  unit 
itself  or  of  one  of  its  aliquot  parts.  We  say  that 
such  a  quantity  is  not  commensurable  (has  not 
a  common  measure)  with  the  unit ;  and  we  call 
the  number  that  exactly  expresses  its  length 
an  incommensurable  number.  Algebraic  symbols 
may  be  used  for  the  number  ( like  V2,  or,  in  the 
case  of  a  circumference,  a  number  like  10  tt), 
but  it  cannot  be  expressed  in  figures. 

The  proofs  of  geometry  for  which  the  method 
60 


THE   METHOD  OF  LIMITS 

of  limits  is  used  all  refer  to  incommensurable 
numbers.  The  task  attempted  in  every  case  is 
to  show  that  computations  from  certain  direct 
measurements  will  give  the  measurement-num- 
ber of  the  quantity  in  question. 

In  general  the  method  of  attack  serves  to  ob- 
scure the  problem.  That  method  is  an  inheritance 
from  Euclid,  in  whose  time  units  were  in  such  a 
confused  state  that  they  could  hardly  be  spoken 
of  as  standard  even  for  a  particular  time  and  in 
a  particular  country ;  and  for  whom  the  number 
system  was  cumbrous  in  the  extreme.  According 
to  that  time-honored  method,  quantities  are  com- 
pared with  each  other  as  if  there  were  no  unit. 
From  that  comparison  we  moderns  deduce,  as  an 
incidental  consequence,  of  great  practical  but  of 
small  theoretical  import,  the  case  where  one  of 
the  quantities  has  fallen  from  its  high  estate  and 
become  a  mere  unit. 

Since  one  of  our  important  aims  in  teaching 
geometry  must  be  to  foster  in  the  pupil's  mind 
the  concept  of  a  series  of  numbers  exactly  corre- 
sponding to  every  series  of  quantities,  we  should 
be  disappointed  if  he  did  not  almost  instinct« 
6i 


TEACHING  MATHEMATICS 

ively  think,  for  example,  of  an  angle  as  a  cer- 
tain number  of  degrees.  He  does,  in  fact.  The 
teacher,  moreover,  will  give  numerical  illustra- 
tions of  his  theorem,  tacitly  assuming  the  very 
truth  he  is  engaged  in  proving.  The  result  is 
that  instead  of  investigating  one  measurement- 
ratio,  that  of  a  measurable  quantity  to  the  unit 
of  that  kind  of  quantity  —  rather  an  abstract 
problem  at  best,  involving  the  fundamental  ide? 
of  number  —  the  pupil  is  really  dealing  in  a 
somewhat  vague  way,  wholly  divorced  from  such 
experience  as  he  may  have  had  with  actual  things, 
with  the  ratio  of  two  numbers  each  of  which  is  it- 
self a  ratio  ( as  of  course  all  measurement  num- 
bers are);  though  the  formal  words  of  the  proof 
do  not  mention  those  numbers  at  all,  and  the 
teacher  ( of  whom  the  text-book  is  a  part )  will 
deny,  in  spite  of  his  numerical  illustrations,  that 
there  are  any  such  numbers  in  sight  until  he 
has  planted  his  foot  on  the  last  step  of  the 
proof. 

No  loss  in  rigor,  and  much  gain  in  clearness, 
would  result  if  we  started  in  every  case  with  the 
unit  as  one  of  the  two  quantities  compared.  For  ex' 
62 


THE  METHOD  OF  LIMITS 

ample,  the  theorem  that  a  central  angle  is  pro- 
portional to  its  intercepted  arc  can  be  stated  thus  : 

The  measurement-number  of  any  central  angle 
is  the  same  as  that  of  its  intercepted  arc ; 

or  perhaps,  to  avoid  confusion  with  the  length- 
number  of  an  arc,  thus  :  — 

The  number  of  degrees  in  an  angle  is  the  same 
as  the  number  of  degrees  in  its  intercepted 
arc. 

The  proof  consists  first  in  defining  a  degree  of 
arc,  and  showing,  by  a  previous  theorem,  that, 
while  it  may  be  of  different  lengths  on  dif- 
ferent circles,  on  the  same  or  equal  circles  a  de- 
gree of  arc  is  congruent  with  every  other  degree 
of  arc.  Consequently  an  angle  of  an  integral 
number  of  degrees  intercepts  an  arc  of  the  same 
number  of  degrees. 

Again  a  tenth  of  a  degree  of  angle  is  one  of 
ten  mutually  congruent  parts  of  a  degree  ;  each 
will  intercept,  according  to  the  same  theorem 
before  referred  to,  one  of  ten  mutually  con- 
gruent parts  of  a  degree  of  arc  ;  and  likewise 
for  other  aliquot  parts  of  a  degree. 

Thus  we  have  established   the   theorem  for 
63 


TEACHING  MATHEMATICS 

all  cases  where  the  number  of  degrees  in  the 
angle  can  be  expressed  in  figures.  If  we  have 
an  angle  in  which  the  number  of  degrees  can- 
not be  so  expressed,  in  other  words,  where  it  is 
incommensurable,  we  must  still  base  our  proof 
upon  degrees  and  fractions  of  a  degree.  We  can 
show  the  existence  of  a  series  of  central  angles, 
each  greater  than  the  one  before  it,  and  each  less 
than  the  angle  we  have  to  measure ;  and  we  can 
also  show  the  existence  of  a  series  of  arcs  each 
greater  than  the  one  before  it,  and  each  less  than 
the  arc  we  have  to  measure  ;  moreover  each  arc 
is  intercepted  by  the  angle  corresponding  to  it  in 
the  other  series,  and  the  arc  and  its  correspond- 
ing angle  have  the  same  measurement-number. 

These  series  can  be  continued  until  the  last 
angle  differs  from  the  angle  to  be  measured  by  an 
angle  less  than  any  specified  subdivision  of  a 
degree;  the  last  arc  will  then  differ  from  the 
arc  to  be  measured  by  an  arc  less  than  the  same 
specified  subdivision  of  a  degree  ;  and  the  mea- 
surement-number, the  number  of  degrees  for 
the  last  angle,  being  the  same,  as  we  have  seen, 
for  the  last  arc  also,  differs  from  either  of  the 
64 


THE  METHOD  OF  LIMITS 

measurement-numbers  sought  by  less  than  the 
specified  subdivision  of  unity  ;  the  measurement- 
number  of  the  arc,  then,  cannot  differ  from  the 
measurement-number  of  the  angle,  because  either 
is,  by  a  definition  to  be  given  later,  the  limit  of 
the  same  sequence  of  numbers. 

This  proof  is  so  important,  and  so  typical  of 
proofs  for  the  measurement  of  quantities  incom- 
mensurable with  the  unit,  that  it  should  be  fully 
illustrated.  Again,  it  must  be  remembered  that 
the  pupil's  conception  of  incommensurable  num- 
bers must  be  developed  by  just  such  proofs ;  it 
follows  then  that  he  will  be  perplexed  not  only 
by  the  proof  itself  but  by  the  question  of  the 
necessity  for  it. 

For  the  first  instance  of  this  kind  of  proof, 
therefore,  it  is  advisable  to  select,  for  illustrative 
purposes  only,  a  quantity  that  actually  has  a  com- 
mensurable measurement-number,  but  one  that 
cannot  be  expressed  in  decimal  notation.  Let  us 
say,  the  arc  intercepted  by  a  central  angle  of 
io|°.  Here  we  can  first  show  that  the  measure- 
ment-number, io|,  is  the  same  for  the  arc  and 
for  the  angle,  as  stated  in  the  theorem.  Then  we 
65 


TEACHING  MATHEMATICS 

can  describe  the  construction  of  the  series  of 
angles,  and  the  series  of  arcs,  with  the  corre- 
sponding series  of  measurement-numbers  as  fol- 
lows :  — 

10.3°,  10.33°,  10.333°,  10-3333°.  and  so  on. 

We  can  show  that  the  successive  angles,  arcs, 
and  measurement-numbers  differ,  respectively, 
from  those  we  are  after  by  less  than  .1°,  .01°, 
.001°,  .0001°,  and  so  on  ;  and  consequently,  even 
if  we  did  not  know  that  the  required  measure- 
ment number  was  10^,  we  could  prove  that  the 
number  for  the  arc  could  not  differ  from  the 
number  for  the  angle  by  any  decimal  fraction 
that  might  be  specified  in  advance,  however 
small  that  fraction  might  be. 

This  kind  of  proof  is  more  welcome  to  the  pu- 
pil, for  one  reason  at  least :  namely,  that  a  standard 
system  of  subdivision  for  the  unit  is  less  vague 
than  "any  convenient  fraction,"  and  recommends 
itself  to  a  healthy  practical  sense.  There  is  also, 
as  I  have  said,  the  postponement  of  this  new 
idea  of  incommensurable  numbers  until  the  ar- 
gument that  establishes  their  reason  for  existence 
is  more  familiar.  Perhaps  I  am  wrong  in  saying 
66 


THE  METHOD  OF  LIMITS 

that  the  idea  should  be  postponed ;  at  least  it 
should  not,  while  new  and  abstruse  itself,  be  made 
a  necessary  part  of  an  entirely  new  kind  of  ar- 
gument, itself  sufficiently  abstruse. 

Precisely  the  same  problem  presents  itself 
when  we  begin  to  deal  with  the  measurement  of 
the  rectangle.  Considering  first  the  strip  of  unit 
width  along  one  base  of  the  rectangle,  we  have  a 
unit  square  standing  on  each  length-unit  of  the 
base,  a  tenth  of  a  unit  square  on  each  tenth  of  a 
length-unit,  and  so  on.  The  theorem  upon  which 
our  argument  is  founded  is  that  two  rectangles 
with  the  bases  and  altitudes  respectively  equal 
are  congruent.  When  the  length-number  of  the 
base  is  incommensurable  we  have  the  same  kind 
of  correspondence  as  before,  this  time  of  a  series 
of  lines,  a  series  of  rectangular  unit-wide  strips 
on  those  lines  as  bases,  and  a  series  of  measure- 
ment numbers.  The  argument  is  identical  with 
that  just  given  for  the  central  angle  and  its  inter- 
cepted arc. 

When  we  come  to  deal  with  the  entire  area  of 
the  rectangle,  the  different  series  of  correspond- 
ing quantities  and  measurement-numbers  is  a 
67 


TEACHING  MATHEMATICS 

little  more  complicated.  Let  us  represent  the 
length-numbers  of  the  sides  of  the  rectangle 
(whether  commensurable  or  not  )  by  a  and  b,  and 
its  area-number  by  5".  We  can  form  a  series  of 
pairs  of  sides,  whose  commensurable  length-num- 
bers are  represented  by  x  and  y,  each  pair  form- 
ing a  rectangle  whose  area  Q  consists  of  ;r  unit- 
wide  strips  each  of  area  y.  We  have  then  for  each 
rectangle  in  the  series  the  equation  Q=xy. 

Now  when  for  example  we  measure  the  sides 
of  the  rectangle  to  tenths  of  the  length-unit,  the 
rectangle  Q  cannot  differ  from  the  rectangle  5 
by  an  amount  so  great  as  the  area  of  a  strip  -J^ 
of  a  unit  wide  running  around  two  sides  of  the 
rectangle  Q.  That  is 

and,  since  ;r  ■<  «  and  y  <^b,  and  also  ^o  "^C  ^ 
S-Q<-h{a  +  b  +  I) 
In  the  same  way  when  we  measure  to  hun- 
dredths, thousandths,  or  any  other  decimal  sub- 
division. For  any  such  subdivision,  say  to  ;/ths, 
we  have 

S-Q<l{a+  b  4-  I) 
This  difference  can  be  made  less  than  any  num- 
68 


THE  METHOD  OF  LIMITS 

ber  specified  in  advance,  however  small,  if  we 
imagine  the  subdivision  of  the  unit  carried  far 
enough. 

In  the  same    way  (;tr  +  ^)   ( j  +  „  )    would 
make  a  number  larger  than  ab ;  so  that 

ab-xy  <^\  {x  +  y  +  i) 
or  as  before 

ab—xy<^l  {a  +  b  +  i) 
Now,  since  Q=xy,  S  and  ab  are  merely  dif- 
ferent symbols  for  the  number,  not  yet  expressed 
in  figures,  which  is  the  limit  of  the  series  of  num- 
bers each  of  which  is  represented  by  Q,  or  hyxy. 
In  other  words, 

S=ab 

The  argument  establishing  the  existence  of 
the  number  tt  is  of  the  same  sort.  If  we  deal 
with  the  area  of  the  circle,  we  obtain  a  series  of 
inscribed  polygons,  with  their  area-numbers,  each 
larger  than  the  preceding ;  each  polygon  is 
smaller  than  the  circle.  We  also  obtain  a  series 
of  circumscribed  polygons,  each  smaller  than  the 
one  preceding,  and  every  one  of  them  larger  than 
the  circle  itself ;  corresponding  is  a  series  of  area' 
numbers  for  these  polygons. 
69 


TEACHING   MATHEMATICS 

The  problem  of  area  is  a  good  one  to  attack 
first,  because  there  is  no  question  about  what 
is  meant  by  the  quantity  to  be  measured.  The 
length  of  the  circumference  is  a  different  matter, 
for  no  part  of  it  can  be  considered  congruent 
with  the  unit  of  length,  or  with  any  fraction  of  it. 
The  latter  problem  may  be  approached  as  fol- 
lows :  — 

If  we  begin,  say,  with  the  perimeter  of  a  reg- 
ular hexagon  inscribed  in  the  circle,  and  then 
obtain  the  perimeters  of  the  regular  inscribed 
polygons  of  12,  24,  48,  96,  etc.,  sides,  succes- 
sively, we  shall  have  a  series  of  length-numbers 
which  we  may  represent  by 

A*  pYl^  Pu>  As)  /%  •    •    •    • 

each  of  which  is  greater  than  the  one  preceding. 
Then  if  we  start  with  the  circumscribed  hex- 
agon, and  repeatedly  double  the  number  of  sides, 
we  can  obtain  another  series  of  length-numbers 
which  we  may  represent  by 

^6.   $^12,   q-ih   ^48.   $^96  •    •     •     • 

each  of  which  is  less  than  the  one  preceding. 

We   have  two  things  to  prove  before  going 
on;  first,  that  any  circumscribed  polygon  has  a 
70 


THE  METHOD  OF  LIMITS 

greater  perimeter  than  any  inscribed  polygon; 
and  second,  that  we  can  find  enough  correspond- 
ing pairs  of  terms  in  the  two  series  so  that  the 
difference  between  the  last  /  and  the  last  q  shall 
be  less  than  any  number  specified  in  advance. 

If  we  now  imagine  the  two  series  of  numbers 
to  be  represented  by  two  series  of  points  on  a 
line,  that  is,  in  the  case  of  a  circle  with  one-inch 
radius,  so  that  OPq=  just  6  inches,  (9^e  =  6.928 
inches ;  P12,  P^x,  Pis  ^^^  so  on  would  be  points 
between  Pq  and  Q^,  and  ^12,  Q2i,  Qa  and  so  on 
would  also  be  points  between  P^  and  Q^. 

In  this  diagram  the  points  /'12,  /*24>  etc.,  re- 
presenting the  numbers /i2»  ^24)  etc.,  would  begin 

Tbis-T?ay  for  0  -^  P04        Q„.       ^« 

'  I      I     I"  III!    I'll 

Fig.  5 

at  Pe  and  succeed  each  other  towards  the  right, 
while  the  Q's  begin  at  Q^  and  succeed  each  other 
towards  the  left.  No  P  can  appear  on  the  right 
of  any  point  Q,  and  no  Q  can  appear  on  the  left 
of  any  point  P.  We  can  make  a  P-Q  pair  of 
points  as  close  to  each  other  as  we  please  by 
71 


TEACHING  MATHEMATICS 

continuing  the  process  of  computing  polygons  of 
double  the  number  of  sides,  but  there  is  always 
the  inexorable  law  that  no  /*- point  can  appear 
in  any  part  of  the  region  in  which  a  Q  has  yet 
appeared,  or  in  which,  by  continuation  of  our 
work,  a  Q  may  hereafter  appear. 

If  we  assume  then,  that  there  is  a  point  L,  to 
the  right  of  all  the  Fs  and  to  the  left  of  all  the 
Qs ;  and  that  there  is  a  corresponding  number 
/,  greater  than  every  one  of  the/'j  and  less  than 
every  one  of  the  q's  ;  then  we  can  call  that  point 
and  its  corresponding  number  the  limits  to  which 
the  points  and  numbers  that  we  have  been  con- 
sidering approach.  It  satisfies  the  definition  of 
a  limit,  as  follows:  — 

A  variable  is  said  to  approach  a  constant  as  a 
limit  when,  no  matter  what  small  number  is 
specified  in  advance,  some  one  of  the  regular 
sequence  of  values  assumed  by  the  variable, 
as  well  as  every  value  thereafter,  differs  from 
the  constant  by  an  amount  less  than  the  small 
number  arbitrarily  specified  in  advance. 

It  has  been  proposed  to  define  the  length  of 
the  circumference  of  a  circle  as  the  limit  of  the 
length  of  the  perimeter  of  an  inscribed  polygon, 

72 


THE  METHOD  OF   LIMITS 

the  number  of  sides  being  doubled  again  and 
again  indefinitely.  This  has  at  least  the  advant- 
age of  being  the  only  definition  possible.  If  it  is 
objected  to  on  the  ground  that  it  is  too  abstruse 
for  this  stage  of  education,  that  is  a  reason  for 
omitting  the  argument  by  limits  from  the  study 
of  the  circle  in  high-school  geometry;  it  is  not  a 
good  reason  for  passing  over  the  definition  of  the 
length  of  a  curved  line  as  if  there  were  no  diffi- 
culty there. 

The  objection  that  we  get  a  different  series  of 
numbers  if  we  begin,  say,  with  a  square  or  a 
pentagon  is  seen  to  be  of  no  moment  when  we 
remember  that  each  of  the  numbers  of  any  such 
series  as 

A»  /sj  /i6.  /32>  A4   •     •    •     • 

must  still  be  less  than  any  q  whatever,  whether 
of  the  series  corresponding  to  this  series  of  /'j, 
or  of  any  other ;  and  consequently  the  limit  / 
must  be  the  limit  of  every  such  series. 


IX 


y^ 


SIMPSON'S  RULE  AND  THE  CURVE  OF  SECTIONS 

Simpson's  Rule  for  plane  areas  is  a  formula  for 

obtaining  the  area  of  a  figure  bounded   by    a 

curv^ed  line  ;  for  exam- 
ple, the  water-line  plan 
of  a  ship.  It  depends 
on  the  area  of  a  double 
strip  such  as  is  shown 
in  this  diagram.  Here 
the  figure  bounded  by 
yiy  2  k,  j'3,  and  the 
curve  is  approximated 
to  by  a  rectangle  and 

two  trapezoids,    each   having  the  breadth  §  k. 

These  figures  have  areas  as  follows :  — 

First  trapezoid  :  |(ji  +  J2)  I  ^  =  I  (ji  +  J'2) 

Rectangle:  (§/^)j2  =  1(272) 

Second  trapezoid  :  h  (72  -^y^)  l^  =  i  ( J2  +  Ja) 
The  entire  polygon,  then,  which  is  intended  to 

74 


/T 

'T^ 

1 
1 

1 

1 
1 
1 
1 

\ 
1 

1 

— •( 
Fig.  6 


Fig.  7 


SIMPSON'S   RULE 

be  an  approximation  to  the  area  of  the  given  fig- 
ure, has  an  area 

S  =  |  (ji  +  4/2  +  Js) 
Generally 

there  are  sev- 
eral  double 

strips,  as  in 

the  diagram 

here  given. 

The  end-ordinate  may  be  zero,  but  takes  its 

place  in  the  formula  just  the  same.  Thus  in  this 

diagram  the  areas  of  the  five  double  strips  are  :  — 
First  double  strip  :    |  (jo  +  4/i  +  J2) 
Second    "        "         |  (/2  +  4/3  +  n) 
Third      "        "         I  (j/4  +  4/5  +/6) 

and  so  on  ;  the  entire  area  being  given  by  the 

formula 

5=1  (j'o  +4/1  +  2/2  +  4J'3  +  2j4  +  475  +  2/ 

+  4f-  +    2/8  +  4J9  +/10) 

The  numbers  i,  4,  2,  4,  2  ....  are  called 
"  Simpson's  multipliers." 

For  a  curve  that  is  not  too  steep  the  results 
from  this  formula  are  very  accurate.  Without  it 
the  pupil  has  no  means  of  handling  any  curve 
75 


TEACHING  MATHEMATICS 

except  the  circle.  With  it  he  has  a  practical  ap. 
plication  of  his  knowledge  of  geometry  and  a 
very  satisfactory  command  over  any  kind  of 
area. 

As  an  example  of  its  use  consider  the  follow- 
ing computation  for  the  area  of  a  circle  20  inches 
in  radius,  by  which  the  value  of  tt  to  6  figures  is 
obtained  with  great  simplicity. 

The  equation 
of  the  circle 
with  radius  20 
and  centre  at  O 
is 

x^  -V  y^  =  400 
From  this,  by 
the  use  of  a  ta- 
ble of  square 
roots,  we  obtain 
the  following 
values  of  J,  for 
values  of  x  for 
every  inch  from 
Fig.  8  o  to   lo  inches. 

We  use  them  to  find  the  ordinates  of  half  a  60° 
76 


SIMPSON'S  RULE 

segment.  From  this  we  can  get  the  six  seg- 
ments to  be  added  to  the  area  of  the  inscribed 
hexagon. 

Subtracting  the  length  OP  =/  from  each 
value  of  y,  we  obtain  the  length  of  the  cor- 
responding ordinate  of  the  segment,  which  is 
to  be  used  in  finding  its  area  by  Simpson's 
Rule. 


X 

r 

y 

y-p        Simps 

o 

400 

20.0000 

2.6795 

X 

I 

399 

19.9750 

2.6545 

4 

2 

396 

19.8997 

2.5792 

2 

3 

391 

^9-im 

2.4532 

4 

4 

384 

19-5959 

2.2754 

2 

S 

375 

19.3649 

2.0444 

4 

6 

364 

19.0788 

1.7583 

2 

7 

351 

18.7350 

1-4145 

4 

8 

336 

18.3303 

1.0098 

2 

9 

319 

17.8606 

0.5401 

4 

lO 

300 

17-3205 

0,0000 

X 

Then,  using  Simpson's  multipliers  as  indicated 
we  have  the  following  products  :  — 

77 


TEACHING  MATHEMATICS 

2  g^ge  The  sum  of  these  products  gives  the 

10.6180       value  of  the  parenthesis,  in  the  for- 

_     „        mula,  which  we  have  to  multiply  by  -. 
9.8128  ^  -^    -'3 

4.5508       In  this  case /^=  I. 

8-1776  The  area  of  this  half-segment,  then, 

3.5166 
6q8o       ^^  ^  (54-3517);  the  six  segments  that 

2.0196       lie  around  the  inscribed  hexagon  will 

2.1604       , 

have  an  area  6  Q)  (54.3517)  which 

0.0000  -^       ^        Jj    /y 

reduces  to  4(54.3517)  square  inches. 

54-3517  'Yhe  area  of  the  hexagon,  ^,  is 

4 
6(40o)(i. 73205)     ,, 

"^ —  — ^  =  6(173.205)  square  mches. 

4 

The  entire   area  of  the   circle,  then,  will    be 
1256.637  square  inches;  then,  since 
40077=1256.637 
77  =  3.14159 

Suppose  the  area  of  the  base  of  a  solid  is  3.1 1 

sq.  in. ;  the  area  of  a  section  parallel  to  the  base 

and  one  inch  above  it,  3.02  sq.  in. ;  of  another 

section  2  inches  above  the  base,  2.86  sq.  in. ;  3 

78 


SIMPSON'S  RULE 


inches  above,  2.63  sq.  in.  ;  4  inches,  2.36  sq.  in. ; 
5  inches,  2.01  sq.  in.  ;  and  so  on. 

These  numbers  could  be  laid  off  as  ordinates, 


3.11 


2.01 


Q         li 


Fig.  9 


on  any  convenient  scale,  as  in  the  diagram. 

If  now  we  suppose  that  a  great  many  other  sec- 
tions are  measured,  say  at  intervals  of  j\  inch, 
or  even  ^^^  inch  or  j^qq  inch,  the  ordinates  laid 
off  for  them  between  those  here  shown,  and  a 
curve  drawn  through  the  tops  of  all  these  ordi- 
nates, the  curve  will  be  a  graph  of  the  areas  of 
horizontal  sections,  the  distances  OP,  PQ,  etc., 
showing  the  distances  between  two  sections. 
Such  a  curve  is  called  a  curve  of  sectional  areas 
(or,  briefly,  a  curve  of  sections  )  for  the  solid. 

I  shall  presently  prove  that  the  area-number 
of  the  curve  of  sections  is  the  same  as  the  vol- 
79 


TEACHING  MATHEMATICS 

ume-number  of  the  solid  for  which  it  is  drawn. 
This  theorem  is  of  use  to  prove  the  "  Principle 
of  Cavalieri,"  namely  :  — 

If  two  solids  have  equivalent  bases,  and  if 
sections  parallel  to  the  bases  and  equally 
distant  from  them  are  equivalent,  then  the 
solids  are  equivalent. 

Two  solids  such  as  those  here  described  would 
have  the  same  curve  of  sections. 

The  volume-number  of  a  right  prism  of  unit 
thickness  (altitude)  is  equal  to  the  area-number 
of  its  base.  For  upon  every  square  unit  in  its 
base  can  be  laid  a  cubic  unit;  on  every  tenth  of 
a  square  unit,  a  tenth  of  a  cubic  unit,  and  so  on. 
All  the  propositions  about  congruent  or  equiva- 
lent plane  figures  can  be  shown  to  be  true  of  the 
right  prisms  of  unit  thickness  standing  upon 
them.  For  the  sake  of  generahty  we  may  substi- 
tute cylinder  for  prism  in  this  theorem,  under- 
standing by  a  right  cylindrical  surface  that  gen- 
erated by  a  line  tracing  out  the  perimeter  of  the 
base  and  remaining  perpendicular  to  its  plane. 

The  area-number  of  the  base  of  this  unit-thick 
cylinder  being  laid  off  as  an  ordinate,  and  the 
80 


SIMPSON'S  RULE 

unit  thickness  being  measured  off  along  OX,  the 
volume  of  the  cylinder  will  be  represented  by 
the  area  of  the  rectangle  thus  constructed.  The 
volume  of  a  cylinder  -^^  of  a  unit  thick  would 
be  represented  by  a  rectangle  ^^  of  a  unit  wide, 
and  so  on. 

If  now  we  take  any  solid  standing  on  a  hor- 
izontal base,  and  divide  it  by  means  of  horizontal 
sections  into  slices,  the  areas  of  the  sections  will 
appear  as  ordinates  of  the  curve  of  sections  at 
the  appropriate  points  on  OX. 

A  right  cylinder  standing  upon  one  of  those 
sections,  and  having  a  thickness  equal  to  the 
thickness  of  the  slice,  would  have  its  volume 
represented  by  the  area  of  a  rectangle  formed 
upon  the  corresponding  ordinate  in  the  curve  of 
sections. 

Let  us  now  suppose  that  the  solid  is  divided 
up  into  very  thin  slices  of  the  same  thickness, 
and  that  upon  the  base  of  each  slice  we  con- 
struct a  right  cylinder  of  the  same  thickness. 
The  total  volume  of  the  whole  series  of  cylinders 
would  differ  from  the  volume  of  the  solid  by  less 
than  a  layer  of  a  certain  definite  thickness  ex- 
8i 


TEACHING  MATHEMATICS 

tending  over  the  whole  lateral  surface  of  the 
solid.  If  we  now  halve  the  thickness  of  the  slices, 
reconstructing  the  cylinders  to  correspond,  the 
necessary  thickness  of  this  layer  would  become 
less  ;  and  by  repeating  the  halving  process  we 
could  finally  arrive  at  a  point  where  the  differ- 
ence between  the  volume-number  of  the  solid 
and  that  of  the  series  of  right  cylinders  would  be 
less  than  any  small  number  that  may  have  been 
specified  in  advance. 

At  the  same  time  the  narrow  rectangles,  con- 
structed as  above  on  the  ordinates  of  the  curve 
of  sections,  have  a  total  area  that  differs  from  the 
area  under  the  curve  by  an  amount  less  than  the 
area  of  a  band  of  a  certain  definite  width  extend- 
ing along  the  upper  boundary.  By  repeating  the 
process  of  halving  slices,  reconstructing  cylinders, 
and  reconstructing  the  rectangles  that  represent 
the  volumes  of  the  cylinders,  we  could  finally 
arrive  at  a  point  where  the  difference  between 
the  area-number  of  the  curve  of  sections  and  that 
of  the  series  of  rectangles  would  be  less  than  any 
small  number  that  may  have  been  specified  in 
advance. 

82 


SIMPSON'S  RULE 

We  shall  thus  have  a  sequence  of  numbers, 
each  of  which  expresses  not  only  the  volume  of 
the  series  of  right  cylinders  but  also  the  area  of 
the  series  of  rectangles  ;  this  sequence  approach- 
ing a  definite  limit,  which  may  or  may  not  be 
commensurable  ( i.e.,  expressible  in  figures) ;  and 
this  limit  being  the  expression  not  only  of  the 
volume  of  the  solid  but  also  of  the  area  of  its 
curve  of  sections.  In  other  words,  the  volume 
of  any  solid  having  a  curve  of  sections  is  equal 
to  the  area  under  that  curve. 

The  Principle  of  Cavalieri  follows  immediately 
from  this  theorem.  It  was  proved  originally  by 
supposing  the  two  solids  to  be  composed  of  very 
thin  but  uniform  layers,  as  of  sheets  of  paper ; 
since  each  layer  in  one  was  equivalent  to  ( con- 
tained as  much  paper  as)  the  corresponding 
layer  in  the  other,  the  total  amount  in  one  was 
the  same  as  in  the  other. 

The  use  of  the  curve  of  sections  leads  at  once 
to  the  so-called  "Prismatoid  Formula"  for  the 
area  of  a  solid.  This  can  be  directly  proved  for 
a  prismatoid,  that  is,  for  a  solid  bounded  by 
planes,  having  all  of  its  vertices  in  two  parallel 
83 


TEACHING  MATHEMATICS 

planes.  It  is  used  also,  as  a  formula  of  approxi- 
mation, for  other  solids,  such  as  those  computed 
for  excavations  and  embankments.  It  consists 
merely  in  the  application  of  Simpson's  Rule  to 
the  section-areas,  on  the  hypothesis  that  they 
are  the  ordinates  of  the  curve  of  sections  ;  for 
the  prismatoid  only  the  mid-section  is  used  in 
addition  to  the  two  bases. 

There  is  need  only  to  mention  the  great  eco- 
nomy in  the  demonstration  of  the  mensuration 
theorems  of  solid  geometry  from  the  early  proof 
of  this  "  principle  "  ;  an  economy  not  merely  of 
actual  effort  in  demonstration,  but  also  of  the 
interest  of  the  pupil,  in  the  presentation  of  one 
comprehensive  method  of  attack  instead  of  a 
number  of  widely  different  methods. 


X 

THE  TEACHER 

Text-books  may  be  written  embodying  these 
or  other  reforms  in  method  or  subject-matter ; 
but  the  success  of  such  projects  must  depend 
upon  the  teacher.  However  well  convinced  the 
teacher  is  of  the  value  of  the  changes  that  he 
wishes  to  effect,  he  cannot  ignore  the  value  that 
exists  in  the  old  ways  ;  nor  can  he  avoid  the  high 
duties  to  which  the  new  times  will  call  him,  and 
from  which  his  propositions  for  reform  will  not 
excuse  him. 

They  ascribe  to  Euclid  a  fable  about  a  mule, 
which  met  a  donkey  at  a  ford,  and  entertained 
him  with  astute  remarks  about  the  sizes  of  their 
respective  burdens.  From  these  remarks  the 
learned,  of  ancient  times,  were  wont  to  infer  the 
load  each  of  these  gossipy  animals  bore.  Ever 
since  that  time  problems  of  a  like  "unpractical" 
character  have  occupied  the  attention  of  stu- 
dents. Even  within  the  memory  of  men  now 
85 


TEACHING  MATHEMATICS 

living,  mathematical  earwigs  have  disported 
themselves  upon  perfectly  rigid  and  unbeliev- 
ably slender  rods,  policemen  having  no  dimen- 
sions have  chased  infinitesimal  culprits  up  math- 
ematical alleyways,  and  we  have  rearranged  the 
soldiers  of  Napoleon  in  phalanxes  that  even 
Xerxes  could  have  seen  were  useless.  Yet  per- 
haps these  problems  were  not  entirely  useless. 
What  did  Apollo  want  to  double  the  size  of  his 
altar  for  ?  The  very  futility  of  the  task,  as  in  the 
quest  of  the  alchemists,  filled  the  world  with 
results  of  great  value. 

Let  us  remember  the  civilization  of  China,  for 
which  confident  philanthropists  would  substitute 
the  ways  of  Europe  or  the  United  States.  It  has 
maintained  itself  for  three  thousand  years :  it 
cannot  be  worthless.  Though  we  have,  to  our 
own  satisfaction,  demolished  the  position  of 
those  who  would  defend  the  established  customs 
of  teaching,  let  us  be  gentle  with  the  poor  old 
world.  There  is  something  eternal  even  in  the 
mistakes  that  we  condemn.  It  is  good  to  be 
enthusiastic  ;  but  to  be  intolerant  is  bad  strategy. 

The  proposition  to  unite  the  different  branches 
86 


THE  TEACHER 

of  school  mathematics  into  one  progressive  sub- 
ject has  a  long  history  of  defeat,  as  the  present 
customs  of  teaching  show.  The  reason  for  that 
defeat  lies  in  the  desire  of  the  community,  as  well 
as  of  the  teachers,  for  distinctly  marked  stages 
of  advancement,  recognizable  successive  tasks, 
for  which  teacher  and  pupils  can  severally  be 
held  responsible,  and  for  which  books  can  be 
ordered  without  too  much  scrutiny. 

The  hope  for  success  in  the  present  wide- 
spread attempt  rests  upon  new  conditions.  For 
one  thing,  teachers  are  better  informed,  less  dis- 
tracted by  demands  to  teach  from  all  parts  of  the 
cyclopaedia,  more  wide  awake  not  only  to  progress 
in  the  art  of  teaching  but  also,  let  us  hope,  to  the 
widening  scope  and  the  beauty  of  the  science  of 
mathematics.  Again,  the  modern  aim,  while  not 
forgetting  that  instruction  is  to  be  sound  and  in 
line  with  later  study,  seeks  also  immediate  effi- 
ciency wherever  possible ;  so  that  the  credentials 
of  progress  are  the  increased  powers  of  the  pupil, 
rather  than  documents  to  show  that  he  "  is  in  " 
or  "  has  been  through  "  the  fields  of  knowledge 
that  bear  the  orthodox  text-book  names. 
87 


TEACHING   MATHEMATICS 

These  considerations  point  to  new  demands 
upon  the  teacher.  The  most  obvious  is  that  he 
can  no  longer  —  any  more  than  can  the  teacher 
of  history  or  French  —  rest  content  with  the 
programme  of  the  text-book  to  which  his  year's 
work  is  married.  The  worded  problems  particu- 
larly he  must  study  out  in  detail,  with  especial 
reference  to  two  things :  first,  the  degree  of  diffi- 
culty in  constructing  the  equation  ;  and  second, 
the  type  of  solution  required  by  that  equation. 
In  geometry  the  teacher  must  not  only  master 
with  great  minuteness  the  logical  relations  and 
the  information  presented  in  the  book  his  pupils 
use,  but  he  must  himself  be  able  to  vary  funda- 
mentally those  logical  relations,  to  classify  and 
reclassify  that  body  of  knowledge,  so  as  to  build 
parts  of  it  from  time  to  time  coherently  about 
the  topics  on  which  he  succeeds  in  arousing  the 
interest  of  the  class. 

More  than  all  else,  and  in  mathematics  more 
than  in  any  other  subject,  the  teacher  should 
have  the  enthusiasm  of  the  achieving  student.  I 
wish  I  could  reproduce  here  the  eloquent  words 
in  which  I  once  heard  a  distinguished  scholar 
88 


THE  TEACHER 

urge,  upon  prospective  teachers  of  mathematics 
in  an  Eastern  university,  the  need  and  the  sure 
reward  of  hard  study  in  lines  not  too  remote 
from  the  field  of  teaching.  Certainly  no  teacher 
of  mathematics  should  rest  content  with  the  bare 
knowledge  of  the  matters  he  is  at  work  upon 
with  his  class  ;  much  of  the  material  to  which  it 
is  directly  preparatory  should  also  be  within  his 
grasp ;  he  should  have  some  knowledge  of  such 
obvious  applications  as  are  known  to  the  machin- 
ist, the  engineer,  the  mariner  —  as  well  as  the 
salesman  and  the  usurer;  and  formal  logic — not 
necessarily  the  time-honored  gabble  of  technical 
terms,  but  the  meat  of  the  subject,  with  some 
idea  of  its  diagrams  and  its  algebra  —  is  almost 
indispensable. 

Not  only  these  things  that  he  must  have  as 
the  weapons  of  his  daily  war,  but  other  treasures 
that  lie  in  great  abundance,  waiting  only  for  re- 
solute endeavor  to  seize  them — treasures  that 
genius  and  incredible  industry  have  heaped  for 
centuries  upon  the  altars  of  wisdom :  these  must 
the  true  teacher  search  for  his  own  adornment. 
Lest  his  mind  become  dulled  in  repeating  stale 
89 


TEACHING  MATHEMATICS 

arguments  to  the  feebler  minds  of  children,  let 
him  whittle  upon  matters  hard  enough  to  test  its 
edge.  Let  him  feel  for  himself  the  triumph,  the 
glow  of  discovery,  that  he  sees  shining  in  the 
eyes  of  those  to  whom  he  is  a. sage.  For  this 
purpose  some  will  choose  one  study,  some  an-' 
other ;  there  is  certainly  variety  enough  to  keep 
conversation  sweet. 


THE    END 


OUTLINE 

I.    THE  MODERN  POINT  OF  VIEW 

1.  The  aim  for  immediate  efficiency I 

2.  Computation  and  self-reliance 3 

3.  Practical  aspects 4 

4.  Definitions  for  the  teacher 5 

5.  Greek  geometry  and  modern  beginners      ....  7 

6.  Algebra  named  for  an  old  Arabic  rule 8 

7.  Purely  symbolic  manipulation 1 1 

II.     THE  ORDER  OF  TOPICS 

1.  Pedantry  in  teaching 13 

2.  Equations  and  numberecl  diagrams 13 

3.  Outline  for  the  first  year 14 

4.  Subjects  for  the  second  year 15 

III.     EQUATIONS  AND  THEIR  USE 

1.  The  ends  to  be  sought 17 

2.  The  subject-matter  of  problems 18 

3.  Positive  terms  only 19 

4.  Negative  terms,  and  the  word  "  algebra"  ....  20 

5.  Geometric  material 22 

IV.     SOME  RULES  OF  THUMB 

1.  Measure  of  precision 23 

2.  What  a  significant  figure  is 23 

3.  The  degree  of  precision  attainable 24 

4.  Reasonable  economy  in  computation 25 

91 


OUTLINE 

5.  The  cultivation  of  self-responsibility 26 

6.  Square  root  of  numbers 26 

7.  The  "guess  and  try  "  method 27 

8.  Its  theory 28 

9.  Its  advantages 28 

10.  Possible  extension  to  cube  root 30 

11.  Algebraic  evolution  to  be  omitted 30 

V.     GEOMETRY   AS   ALGEBRAIC   MATERIAL 

1.  The  facts  available 32 

2.  The  rectangle  rule  fundamental 32 

3.  Rearrangement  necessary 33 

4.  The  stripe  idea 33 

5.  Proof  for  the  rectangle  rule 34 

6.  The  logical  scheme  of  theorems 34 

7.  Appropriate  notation 35 

8.  Illustration  by  the  Pythagorean  Theorem     ...  38 

9.  The  use  of  measurable  angles 40 

10.  Three-figure  table  of  tangents 40 

11.  '•  Practical  problems"  again 40 

VI.    THE  GRAPHICAL  METHOD 

1.  Importance  and  popular  use 41 

2.  Application  to  elimination 42 

3.  The  locus  idea 43 

4.  The  graphical  check 43 

5.  The  circle-equation 44 

6.  A  source  of  elimination  problems 45 

7.  The  standard  parabola 47 

8.  Check  for  one-letter  quadratics 47 

9.  Things  to  be  avoided 47 

92 


OUTLINE 

VII.    THE  BASES  OF    PROOF   IN  GEOMETRY 

1.  The  idea  of  motion  in  Euclid 49 

2.  Logical  system  desirable 50 

3.  Warrant  for  the  early  proofs 50 

4.  Characteristic  motions  of  plane  figures     .     .     .     .51 

5.  Comparison  with  cylinders  and  spheres    ....  52 

6.  Cyclic  order  of  parts S3 

7.  Illustration  by  Euclid  I,  4 54 

8.  Congruence,  symmetry,  and  mensuration      ...  57 

9.  The  early  theorems 57 

ID.  Problems  of  construction 58 

II.  Inequality  theorems  to  be  postponed 59 

VIII.    THE  METHOD  OF  LIMITS 

1.  Incommensurable  numbers 60 

2.  Computations  from  direct  measurement  ....  61 

3.  Our  cumbersome  methods  inherited 61 

4.  The  measurement  of  an  angle 63 

5.  Type  of  the  "limit"  proof 65 

6.  Division  of  difficulties  for  the  pupil      .     .     .     ,     .  65 

7.  The  measurement  of  the  rectangle 67 

8.  The  argument  in  regard  to  tt 69 

9.  Graphical  illustration  of  a  limit 7' 

10.  Definition  of  a  limit 72 

1 1.  Definition  of  the  length  of  the  circumference    .     .  72 

IX.     SIMPSON'S  RULE  AND  THE  CURVE 
OF   SECTIONS 

1.  Simpson's  rule  for  plane  areas 72 

2.  The  double  strip  polygon 74 

3.  Simpson's  multipliers        75 

93 


OUTLINE 

4.  Application  to  the  evaluation  of  w 76 

5.  Curve  of  sectional  areas 78 

6.  The  principle  of  Cavalieri 80 

7.  Volume  of  a  right  cylinder 80 

8.  Generalization  of  volume-mensuration        ....  84 

X.     THE  TEACHER 

1.  The  need  of  tolerance 85 

2.  New  hopes  of  progress 87 

3.  New  tasks  for  the  teacher 88 

The  teacher  a  student      ...         88 


4- 


This  book  is  DUE  on  the  last  date  stamped  below 


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